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This document discusses hyperbolas, ellipses, and canonical equations in analytic geometry. It provides definitions and parameters for hyperbolas and ellipses, including centers, vertices, foci, axes, and asymptotes. Examples are given of determining parameters from equations and graphing conic sections. The document emphasizes using canonical equations to solve problems involving hyperbolas and ellipses. It also outlines steps for solving problems using analytic geometry, such as identifying the relevant figure and equations.

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Actividad colaborativa 551108 20

This document provides information about ellipses, hyperbolas, parabolas, and circles. It defines key elements of each curve such as foci, vertices, axes, and directrix. It also presents the standard equation for each curve in both canonical form (centered at the origin) and general form (shifted center). Examples are given of shifting the coordinates to obtain equations for non-canonical curves.

Conic section

“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.

Conic sections

This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.

10.1 Parabolas

Identify the vertex, directrix, focus, and axis of a parabola.
Write the equation of a parabola in vertex form.

Plano numérico

The document discusses key concepts about the Cartesian coordinate plane including:
- The coordinate plane consists of two perpendicular number lines that intersect at the origin point.
- The coordinate plane is used to describe the position of points and analyze geometric shapes like circles, parabolas, ellipses, and hyperbolas.
- Formulas are provided to calculate the distance between two points on the coordinate plane and to find the midpoint of a line segment between two points.
- Equations are given for circles, parabolas, ellipses, and hyperbolas centered at various positions on the coordinate plane.
- An example problem is included to find the coordinates of points and distances between points on a coordinate plane diagram.

Lecture #6 analytic geometry

The document discusses parabolas including their parts, graphs, and equations. It defines a parabola as the locus of points where the distance to the focus is equal to the distance to the directrix. The parts of a parabola include the vertex, focus, directrix, axis of symmetry, and latus rectum. The document outlines the graphs of parabolas with the vertex at the origin or at a point (h,k), and opening in different directions. It notes equations will be provided for parabolas with the vertex at the origin or (h,k), but does not show the actual equations.

10.2 Ellipses

Identify the parts of an ellipse
Sketch the graph of an ellipse
Write the equation of an ellipse
Calculate the eccentricity of an ellipse

Coordinate geometry

The document discusses coordinate geometry and the Cartesian coordinate system. It describes how René Descartes proposed using an ordered pair of numbers to describe the position of points on a plane. This allows curves and lines to be described through algebraic equations, linking algebra and geometry. The coordinate plane is defined by perpendicular x and y axes that intersect at the origin. Points on the plane are located using their coordinates (x, y), marking their distance from the two axes. The plane is divided into four quadrants by the intersecting axes.

Actividad colaborativa 551108 20

This document provides information about ellipses, hyperbolas, parabolas, and circles. It defines key elements of each curve such as foci, vertices, axes, and directrix. It also presents the standard equation for each curve in both canonical form (centered at the origin) and general form (shifted center). Examples are given of shifting the coordinates to obtain equations for non-canonical curves.

Conic section

“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.

Conic sections

This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.

10.1 Parabolas

Identify the vertex, directrix, focus, and axis of a parabola.
Write the equation of a parabola in vertex form.

Plano numérico

The document discusses key concepts about the Cartesian coordinate plane including:
- The coordinate plane consists of two perpendicular number lines that intersect at the origin point.
- The coordinate plane is used to describe the position of points and analyze geometric shapes like circles, parabolas, ellipses, and hyperbolas.
- Formulas are provided to calculate the distance between two points on the coordinate plane and to find the midpoint of a line segment between two points.
- Equations are given for circles, parabolas, ellipses, and hyperbolas centered at various positions on the coordinate plane.
- An example problem is included to find the coordinates of points and distances between points on a coordinate plane diagram.

Lecture #6 analytic geometry

The document discusses parabolas including their parts, graphs, and equations. It defines a parabola as the locus of points where the distance to the focus is equal to the distance to the directrix. The parts of a parabola include the vertex, focus, directrix, axis of symmetry, and latus rectum. The document outlines the graphs of parabolas with the vertex at the origin or at a point (h,k), and opening in different directions. It notes equations will be provided for parabolas with the vertex at the origin or (h,k), but does not show the actual equations.

10.2 Ellipses

Identify the parts of an ellipse
Sketch the graph of an ellipse
Write the equation of an ellipse
Calculate the eccentricity of an ellipse

Coordinate geometry

The document discusses coordinate geometry and the Cartesian coordinate system. It describes how René Descartes proposed using an ordered pair of numbers to describe the position of points on a plane. This allows curves and lines to be described through algebraic equations, linking algebra and geometry. The coordinate plane is defined by perpendicular x and y axes that intersect at the origin. Points on the plane are located using their coordinates (x, y), marking their distance from the two axes. The plane is divided into four quadrants by the intersecting axes.

Chapter 7.2 parabola

The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.

Plano cartesiano

The document defines key concepts in analytic geometry including:
- The Cartesian plane as two perpendicular number lines intersecting at the origin.
- Equations for circles, ellipses, parabolas and hyperbolas in Cartesian coordinates.
- How to find the midpoint between two points in the plane by taking the average of the x- and y-coordinates.
- Formulas for calculating the distance between two points using the Pythagorean theorem.

Sistemas de coordenadas

The document discusses different types of coordinate systems, including Cartesian and polar coordinates. It provides definitions and examples of transforming coordinates between the Cartesian and polar systems. The key steps for transformation involve using trigonometric functions like sine, cosine, and inverse tangent. Translating and rotating coordinate axes is also covered, with equations provided for finding new coordinates after such transformations. Examples of graphs in polar coordinates include a parabola and circle.

EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIX

What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths

history of conics

- The conic sections (circles, ellipses, parabolas, and hyperbolas) have been studied for over 2000 years, with Apollonius of Perga making major contributions in the 2nd century BC by rigorously studying them and applying the work to astronomy.
- Apollonius was the first to note that conic sections can be constructed by cutting a circular cone with a plane in different ways: a circle from a perpendicular cut, a parabola from a parallel cut, a hyperbola from a cut through both parts of the cone, and an ellipse from a cut through one part of the cone not parallel to its side.
- Conic sections can be represented by a general second

Coordinate goemetry

Coordinate geometry represents points on a number line with real numbers. The number line places positive numbers to the right of zero and negative numbers to the left.
The Cartesian coordinate system uniquely determines points in two or three dimensional space using perpendicular x and y axes intersecting at the origin. René Descartes developed this system, linking algebra and geometry by describing curves and lines with equations.
The coordinate plane has perpendicular x and y axes. Points are located using their x and y coordinates, which represent horizontal and vertical distances from the origin. The plane is divided into four quadrants numbered counter-clockwise from the top right. Ordered pairs notation (x,y) specifies a point's location by listing the x coordinate first.

Conic Section

MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.

Coordinate geometry

Coordinate geometry describes the position of points on a plane using an ordered pair of numbers (x, y). It was developed by French mathematician René Descartes in the 1600s. The system uses two perpendicular axes (the x-axis and y-axis) that intersect at the origin point (0,0). Values to the right of the x-axis and above the y-axis are positive, while values to the left and below are negative. The plane is divided into four quadrants by these axes.

parabola class 12

This document discusses parabolas, including their key features like the vertex, focus, directrix, and axis of symmetry. It provides examples of how to graph parabolas given their standard form equations, both for parabolas with vertices at the origin and for parabolas with other vertices. It also shows how to write the standard form equation of a parabola when given its focus and directrix.

Conic Sections

Conics are plane curves formed by cutting a double right circular cone with a plane. The type of conic section depends on the angle of the cutting plane to the cone's axis: a circle for perpendicular, ellipse for non-perpendicular, parabola for parallel to the edge, and hyperbola for intersecting both cones. A parabola is the set of points equidistant from a focus point and directrix line, with the vertex halfway between them. The latus rectum connects the endpoints equidistant from the focus, determining how wide the parabola opens. The four types of parabolas depend on the axis and opening direction.

Parabola

The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.

4.1 stem hyperbolas

Hyperbolas are defined by the difference between distances to two fixed points called foci. A hyperbola consists of all points where this difference is a constant. It has two branches, two vertices, and two asymptotes which are the diagonals of an invisible box defined by the hyperbola's x-radius and y-radius. To graph a hyperbola, one puts its equation into standard form to determine the center, radii, and direction of opening, then draws the corresponding box and curves.

Lecture #5 analytic geometry

This document discusses the equations of circles, including the equation of a circle with its center at the origin and the equation of a circle with its center located at a point (h,k). It defines a circle as the set of all points equidistant from a fixed center point, with the distance to the center being the radius. The document provides information to graph circles based on their equations.

Lecture co2 math 21-1

This document provides information about circles and conic sections. It begins with an overview of circles, including definitions of key terms like radius, diameter, chord, and equations of circles given the center and radius or three points. It then covers conic sections, defining ellipses, parabolas and hyperbolas based on eccentricity. Equations of various conic sections are derived based on the location of foci, directrix, vertex and other geometric properties. Sample problems are provided to demonstrate solving problems involving different geometric configurations of circles and conic sections.

Maths project

This document summarizes different conic sections including the parabola, ellipse, and hyperbola. It provides the definitions and key properties of each shape. For parabolas, it describes that any point is at an equal distance from the focus and directrix, and provides the standard equation of y2 = 4ax. For ellipses, it defines them as points whose sum of the distances to two fixed points is a constant, and gives the standard equation of x2/a2 + y2/b2 = 1. For hyperbolas, it describes them as points where the ratio of the distances to the focus and directrix are constant, and provides the standard equation of x2/a2 - y

Coordinate geometry

Coordinate geometry
easy to learn points
class 9 cbse course ncert textbook
Quiz
Maths trivia quiz

Conic Section

This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations and key characteristics for each conic section with varying positions of the vertex. Circles are defined by a center point and radius. Parabolas are defined by a focus, directrix, and vertex. Ellipses are defined by two foci and the sum of distances to these points. Hyperbolas are defined by two foci and the difference of distances to these points. Examples of each conic section in architecture and acoustics are also given.

PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C

This document introduces coordinate geometry and the Cartesian plane. It explains that René Descartes developed a method to describe the position of a point in a plane using two perpendicular lines as axes. Any point can be located using its distance from these intersecting x- and y-axes, known as the point's coordinates. The plane is divided into four quadrants by the axes, and examples are provided to demonstrate how to locate a point using its coordinates.

10.4 Summary of the Conic Sections

Identify a conic section from the characteristics of it equation.
Identify a conic section from its general form equation.

Conic sections and introduction to circles

Conic sections are shapes that result from slicing a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Circles can be defined by the general formula x^2 + y^2 = r^2, where all points are a distance r from the center. The center and radius of a circle can be determined by shifting the circle and setting the x and y components to 0.

Plano Numérico

The document describes key concepts related to the Cartesian plane including:
- The Cartesian plane consists of two perpendicular axes (x and y) intersecting at the origin point.
- Points on the plane are represented as ordered pairs (x,y).
- The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula d = √(x2 - x1)2 + (y2 - y1)2.
- Circles, parabolas, ellipses, and hyperbolas are examples of curves that can be represented on the Cartesian plane using algebraic equations. Their properties and equations are discussed.

Unidad 3 paso4pensamiento geométrico y analitico

Presentación de características de los elementos y temáticas de la Unidad 3: Pensamiento Geométrico y Analítico

Chapter 7.2 parabola

The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.

Plano cartesiano

The document defines key concepts in analytic geometry including:
- The Cartesian plane as two perpendicular number lines intersecting at the origin.
- Equations for circles, ellipses, parabolas and hyperbolas in Cartesian coordinates.
- How to find the midpoint between two points in the plane by taking the average of the x- and y-coordinates.
- Formulas for calculating the distance between two points using the Pythagorean theorem.

Sistemas de coordenadas

The document discusses different types of coordinate systems, including Cartesian and polar coordinates. It provides definitions and examples of transforming coordinates between the Cartesian and polar systems. The key steps for transformation involve using trigonometric functions like sine, cosine, and inverse tangent. Translating and rotating coordinate axes is also covered, with equations provided for finding new coordinates after such transformations. Examples of graphs in polar coordinates include a parabola and circle.

EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIX

What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths

history of conics

- The conic sections (circles, ellipses, parabolas, and hyperbolas) have been studied for over 2000 years, with Apollonius of Perga making major contributions in the 2nd century BC by rigorously studying them and applying the work to astronomy.
- Apollonius was the first to note that conic sections can be constructed by cutting a circular cone with a plane in different ways: a circle from a perpendicular cut, a parabola from a parallel cut, a hyperbola from a cut through both parts of the cone, and an ellipse from a cut through one part of the cone not parallel to its side.
- Conic sections can be represented by a general second

Coordinate goemetry

Coordinate geometry represents points on a number line with real numbers. The number line places positive numbers to the right of zero and negative numbers to the left.
The Cartesian coordinate system uniquely determines points in two or three dimensional space using perpendicular x and y axes intersecting at the origin. René Descartes developed this system, linking algebra and geometry by describing curves and lines with equations.
The coordinate plane has perpendicular x and y axes. Points are located using their x and y coordinates, which represent horizontal and vertical distances from the origin. The plane is divided into four quadrants numbered counter-clockwise from the top right. Ordered pairs notation (x,y) specifies a point's location by listing the x coordinate first.

Conic Section

MS Report, When we talked about the conic section it involves a double-napped cone and a plane. If a plane intersects a double right circular cone, we get two-dimensional curves of different types. These curves are what we called the conic section.

Coordinate geometry

Coordinate geometry describes the position of points on a plane using an ordered pair of numbers (x, y). It was developed by French mathematician René Descartes in the 1600s. The system uses two perpendicular axes (the x-axis and y-axis) that intersect at the origin point (0,0). Values to the right of the x-axis and above the y-axis are positive, while values to the left and below are negative. The plane is divided into four quadrants by these axes.

parabola class 12

This document discusses parabolas, including their key features like the vertex, focus, directrix, and axis of symmetry. It provides examples of how to graph parabolas given their standard form equations, both for parabolas with vertices at the origin and for parabolas with other vertices. It also shows how to write the standard form equation of a parabola when given its focus and directrix.

Conic Sections

Conics are plane curves formed by cutting a double right circular cone with a plane. The type of conic section depends on the angle of the cutting plane to the cone's axis: a circle for perpendicular, ellipse for non-perpendicular, parabola for parallel to the edge, and hyperbola for intersecting both cones. A parabola is the set of points equidistant from a focus point and directrix line, with the vertex halfway between them. The latus rectum connects the endpoints equidistant from the focus, determining how wide the parabola opens. The four types of parabolas depend on the axis and opening direction.

Parabola

The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.

4.1 stem hyperbolas

Hyperbolas are defined by the difference between distances to two fixed points called foci. A hyperbola consists of all points where this difference is a constant. It has two branches, two vertices, and two asymptotes which are the diagonals of an invisible box defined by the hyperbola's x-radius and y-radius. To graph a hyperbola, one puts its equation into standard form to determine the center, radii, and direction of opening, then draws the corresponding box and curves.

Lecture #5 analytic geometry

This document discusses the equations of circles, including the equation of a circle with its center at the origin and the equation of a circle with its center located at a point (h,k). It defines a circle as the set of all points equidistant from a fixed center point, with the distance to the center being the radius. The document provides information to graph circles based on their equations.

Lecture co2 math 21-1

This document provides information about circles and conic sections. It begins with an overview of circles, including definitions of key terms like radius, diameter, chord, and equations of circles given the center and radius or three points. It then covers conic sections, defining ellipses, parabolas and hyperbolas based on eccentricity. Equations of various conic sections are derived based on the location of foci, directrix, vertex and other geometric properties. Sample problems are provided to demonstrate solving problems involving different geometric configurations of circles and conic sections.

Maths project

This document summarizes different conic sections including the parabola, ellipse, and hyperbola. It provides the definitions and key properties of each shape. For parabolas, it describes that any point is at an equal distance from the focus and directrix, and provides the standard equation of y2 = 4ax. For ellipses, it defines them as points whose sum of the distances to two fixed points is a constant, and gives the standard equation of x2/a2 + y2/b2 = 1. For hyperbolas, it describes them as points where the ratio of the distances to the focus and directrix are constant, and provides the standard equation of x2/a2 - y

Coordinate geometry

Coordinate geometry
easy to learn points
class 9 cbse course ncert textbook
Quiz
Maths trivia quiz

Conic Section

This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations and key characteristics for each conic section with varying positions of the vertex. Circles are defined by a center point and radius. Parabolas are defined by a focus, directrix, and vertex. Ellipses are defined by two foci and the sum of distances to these points. Hyperbolas are defined by two foci and the difference of distances to these points. Examples of each conic section in architecture and acoustics are also given.

PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C

This document introduces coordinate geometry and the Cartesian plane. It explains that René Descartes developed a method to describe the position of a point in a plane using two perpendicular lines as axes. Any point can be located using its distance from these intersecting x- and y-axes, known as the point's coordinates. The plane is divided into four quadrants by the axes, and examples are provided to demonstrate how to locate a point using its coordinates.

10.4 Summary of the Conic Sections

Identify a conic section from the characteristics of it equation.
Identify a conic section from its general form equation.

Conic sections and introduction to circles

Conic sections are shapes that result from slicing a cone with a plane. There are four types of conic sections: circles, ellipses, parabolas, and hyperbolas. Circles can be defined by the general formula x^2 + y^2 = r^2, where all points are a distance r from the center. The center and radius of a circle can be determined by shifting the circle and setting the x and y components to 0.

Chapter 7.2 parabola

Chapter 7.2 parabola

Plano cartesiano

Plano cartesiano

Sistemas de coordenadas

Sistemas de coordenadas

EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIX

EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIX

history of conics

history of conics

Coordinate goemetry

Coordinate goemetry

Conic Section

Conic Section

Coordinate geometry

Coordinate geometry

parabola class 12

parabola class 12

Conic Sections

Conic Sections

Parabola

Parabola

4.1 stem hyperbolas

4.1 stem hyperbolas

Lecture #5 analytic geometry

Lecture #5 analytic geometry

Lecture co2 math 21-1

Lecture co2 math 21-1

Maths project

Maths project

Coordinate geometry

Coordinate geometry

Conic Section

Conic Section

PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C

PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C

10.4 Summary of the Conic Sections

10.4 Summary of the Conic Sections

Conic sections and introduction to circles

Conic sections and introduction to circles

Plano Numérico

The document describes key concepts related to the Cartesian plane including:
- The Cartesian plane consists of two perpendicular axes (x and y) intersecting at the origin point.
- Points on the plane are represented as ordered pairs (x,y).
- The distance between two points P1(x1,y1) and P2(x2,y2) is given by the formula d = √(x2 - x1)2 + (y2 - y1)2.
- Circles, parabolas, ellipses, and hyperbolas are examples of curves that can be represented on the Cartesian plane using algebraic equations. Their properties and equations are discussed.

Unidad 3 paso4pensamiento geométrico y analitico

Presentación de características de los elementos y temáticas de la Unidad 3: Pensamiento Geométrico y Analítico

Paso 4_Álgebra, trigonometría y Geometría Analítica

The document discusses different types of conic sections including circles, ellipses, hyperbolas, and parabolas. It provides the general equation for conic sections and the conditions to determine which type of conic section is represented based on the values of certain coefficients in the equation. It then gives the standard forms of the equations for each type of conic section and discusses some of their defining geometric properties.

Unidad 3: Pensamiento analítico y geométrico

This document provides an overview of the topics covered in Unit 3 on geometric and analytical thinking, including hyperbolas, ellipses, circles, parabolas, graphs of conic sections, equations of conic sections, and analytical geometry. It discusses the key elements and properties of each conic section. It also presents 4 example problems applying these concepts, such as finding the standard form of an ellipse given its vertices.

Plano cartesiano

The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane is formed by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Any point P on the plane can be located using its coordinates (x,y) which indicate the point's position along the x and y axes.
- The distance between two points P1(x1,y1) and P2(x2,y2) can be calculated using the distance formula.
- Key curves that can be represented on the Cartesian plane include lines, circles, parabolas, ellipses, and hyperbolas through their defining equations.

Tarea 4

This document provides an overview of analytic geometry concepts including:
- The Cartesian plane and using coordinates to locate points
- Equations for lines including vertical, horizontal, and oblique lines
- Geometric loci such as the midpoint and bisector
- Equations for circles, parabolas, ellipses, and hyperbolas
It includes examples of finding equations that model geometric shapes on the Cartesian plane.

Analisis unidad 3

The document discusses geometric and analytical thinking. It begins by defining analytical geometry as the science that combines algebra and geometry to describe geometric figures from both algebraic and geometric viewpoints. It then discusses how analytical geometry originated with René Descartes' use of the Cartesian plane. Several geometric figures are then analyzed, including lines, circles, ellipses, and parabolas. Their key parameters and equations are defined. In particular, it provides the canonical equations for circles, ellipses, and parabolas, and discusses topics like slope and parallelism for lines.

Plano numerico

This document defines and explains key concepts in analytic geometry including:
- The Cartesian plane consisting of perpendicular x and y axes intersecting at the origin.
- Distances between points on the plane and formulas to calculate distances.
- Midpoint of a segment and properties of circles like radius, diameter, and equations of circles.
- Elements and equations of parabolas, ellipses, and hyperbolas including vertices, foci, axes, and canonical forms.
- René Descartes is credited with developing analytic geometry which uses the Cartesian plane.

Plano numerico

The document discusses the Cartesian plane and some of its key elements and uses in geometry. It defines the Cartesian plane as two perpendicular number lines that intersect at an origin point. It describes the axes, quadrants, coordinates, and how geometric shapes like circles and parabolas can be analyzed mathematically using the Cartesian plane. Circles are defined by a center point and radius, and their equations in the Cartesian plane are provided. Properties of parabolas and hyperbolas such as their foci, vertices, and equations are also outlined.

Plano numerico

The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane uses two perpendicular number lines (horizontal and vertical) that intersect at the origin point to describe the position of any other point in the plane using coordinates.
- The distance formula can be used to calculate the distance between any two points given their coordinates.
- Important equations are discussed for lines, circles, ellipses, hyperbolas, and parabolas including their standard/canonical forms in the Cartesian plane.
- Methods for graphically representing these conic sections based on their equations are also covered.

Plano numerico

Plano numérico, punto medio, distancia entre dos puntos, secciones cónicas (ecuaciones, representación gráfica y trazado)

Question 1

The document provides instructions for solving a multi-step math problem to find the coordinates of a bomb. It explains that the clue is the equation of a hyperbola, which needs to be graphed to find the negatively sloped asymptote. Taking the reciprocal of this equation results in another graph whose coordinates at x=1 are the location of the next clue: (1, -4).

Plano numerico ana wyatt

The document describes different types of curves and surfaces that can be represented on the Cartesian plane, along with methods for plotting them. It discusses plotting points, lines, circles, parabolas, ellipses, hyperbolas, and their geometric definitions and properties. Various techniques are provided for tracing each curve type given certain parameters like foci, vertices, directors, etc. The Cartesian plane is established as a useful system for locating points and calculating distances between them.

Plano numerico

This document discusses several topics in geometry including:
- The distance formula to calculate the distance between two points given their coordinates.
- The midpoint formula to find the midpoint of a line segment given the coordinates of the endpoints.
- Equations of lines and circles.
- How to draw an arc or full circle passing through three non-collinear points.
- Definitions and properties of parabolas, hyperbolas, ellipses, and conic sections - curves formed by intersecting a cone with a plane.
- How to represent the equations of conic sections graphically.

Plano Numerico

The document defines and explains key concepts related to coordinate geometry including:
- The Cartesian plane and coordinate systems
- Distances between points in the plane
- Midpoint of a segment
- Common conic sections like circles, parabolas, ellipses, and hyperbolas
- Their defining properties and equations

Plano cartesiano iliaiza gomez

The document discusses the Cartesian plane and its key elements. It describes how René Descartes originated the Cartesian plane by constructing two perpendicular number lines intersecting at a point. The document then defines the key parts of the Cartesian plane, including the x and y axes, quadrants, coordinates, and origin point. It also provides equations for circles, ellipses, hyperbolas, and parabolas in the Cartesian plane.

Plano cartesiano iliaiza gomez

The document discusses the Cartesian plane and its key elements. It describes how René Descartes originated the Cartesian plane by constructing two perpendicular number lines intersecting at an origin point. The document then defines the key parts of the Cartesian plane, including the x and y axes, quadrants, coordinates, and how distance between points is calculated. It also provides the equations for circles, ellipses, hyperbolas, and parabolas - the main conic sections represented using the Cartesian plane.

ellipse

An ellipse is a closed curve where the sum of the distances from two fixed points (foci) to any point on the curve is a constant. It can be defined parametrically using angles or implicitly as a second-degree equation. Key properties include the major and minor axes, which are lines of symmetry, and the eccentricity, which is a measure of how non-circular the ellipse is. The area of an ellipse is πab and its circumference can be calculated using elliptic integrals.

Plano numerico / Matematica

1) An ellipse is a plane curve such that the sum of the distances from two fixed points (the foci) is constant.
2) It can be defined as the intersection of a cone by a plane that is not parallel to the central axis and at an angle greater than the generator.
3) The standard equation of an ellipse centered at the origin is x2/a2 + y2/b2 = 1, where a and b are the semi-major and semi-minor axes.

Precal 3-4.pptx

This document provides information about ellipses and hyperbolas. It defines ellipses as sets of points where the sum of the distances to two fixed foci is a constant. Hyperbolas are defined as sets of points where the difference of distances to two foci is a constant. The document gives equations and properties of ellipses and hyperbolas like foci, vertices, axes, asymptotes. It includes examples of finding standard forms of equations and graphing ellipses and hyperbolas given properties.

Plano Numérico

Plano Numérico

Unidad 3 paso4pensamiento geométrico y analitico

Unidad 3 paso4pensamiento geométrico y analitico

Paso 4_Álgebra, trigonometría y Geometría Analítica

Paso 4_Álgebra, trigonometría y Geometría Analítica

Unidad 3: Pensamiento analítico y geométrico

Unidad 3: Pensamiento analítico y geométrico

Plano cartesiano

Plano cartesiano

Tarea 4

Tarea 4

Analisis unidad 3

Analisis unidad 3

Plano numerico

Plano numerico

Plano numerico

Plano numerico

Plano numerico

Plano numerico

Plano numerico

Plano numerico

Question 1

Question 1

Plano numerico ana wyatt

Plano numerico ana wyatt

Plano numerico

Plano numerico

Plano Numerico

Plano Numerico

Plano cartesiano iliaiza gomez

Plano cartesiano iliaiza gomez

Plano cartesiano iliaiza gomez

Plano cartesiano iliaiza gomez

ellipse

ellipse

Plano numerico / Matematica

Plano numerico / Matematica

Precal 3-4.pptx

Precal 3-4.pptx

clinical examination of hip joint (1).pdf

described clinical examination all orthopeadic conditions .

C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx

C1 Rubenstein

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Constructing Your Course Container for Effective Communication

Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.

spot a liar (Haiqa 146).pptx Technical writhing and presentation skills

sample presentation

Gender and Mental Health - Counselling and Family Therapy Applications and In...

A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!

Hindi varnamala | hindi alphabet PPT.pdf

हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com

Bed Making ( Introduction, Purpose, Types, Articles, Scientific principles, N...

Topic : Bed making
Subject : Nursing Foundation

คำศัพท์ คำพื้นฐานการอ่าน ภาษาอังกฤษ ระดับชั้น ม.1

คำศัพท์เบื้องต้นสำหรับอ่าน ของนักเรียนชั้น ม.1

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...

math operations ued in python and all used

used to math operaions

LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP

This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.

BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...

BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...Nguyen Thanh Tu Collection

https://app.box.com/s/qhtvq32h4ybf9t49ku85x0n3xl4jhr15Main Java[All of the Base Concepts}.docx

This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.

Chapter wise All Notes of First year Basic Civil Engineering.pptx

Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1

Solutons Maths Escape Room Spatial .pptx

Solutions of Puzzles of Mathematics Escape Room Game in Spatial.io

Film vocab for eal 3 students: Australia the movie

film vocab esl

A Independência da América Espanhola LAPBOOK.pdf

Lapbook sobre independência da América Espanhola.

What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...

What is Digital Literacy? A guest blog from Andy McLaughlin, University of Aberdeen

clinical examination of hip joint (1).pdf

clinical examination of hip joint (1).pdf

C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx

C1 Rubenstein AP HuG xxxxxxxxxxxxxx.pptx

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Pengantar Penggunaan Flutter - Dart programming language1.pptx

Constructing Your Course Container for Effective Communication

Constructing Your Course Container for Effective Communication

spot a liar (Haiqa 146).pptx Technical writhing and presentation skills

spot a liar (Haiqa 146).pptx Technical writhing and presentation skills

Gender and Mental Health - Counselling and Family Therapy Applications and In...

Gender and Mental Health - Counselling and Family Therapy Applications and In...

Hindi varnamala | hindi alphabet PPT.pdf

Hindi varnamala | hindi alphabet PPT.pdf

Bed Making ( Introduction, Purpose, Types, Articles, Scientific principles, N...

Bed Making ( Introduction, Purpose, Types, Articles, Scientific principles, N...

คำศัพท์ คำพื้นฐานการอ่าน ภาษาอังกฤษ ระดับชั้น ม.1

คำศัพท์ คำพื้นฐานการอ่าน ภาษาอังกฤษ ระดับชั้น ม.1

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf

math operations ued in python and all used

math operations ued in python and all used

LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP

LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UP

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum

BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...

BÀI TẬP DẠY THÊM TIẾNG ANH LỚP 7 CẢ NĂM FRIENDS PLUS SÁCH CHÂN TRỜI SÁNG TẠO ...

Main Java[All of the Base Concepts}.docx

Main Java[All of the Base Concepts}.docx

Chapter wise All Notes of First year Basic Civil Engineering.pptx

Chapter wise All Notes of First year Basic Civil Engineering.pptx

Solutons Maths Escape Room Spatial .pptx

Solutons Maths Escape Room Spatial .pptx

Film vocab for eal 3 students: Australia the movie

Film vocab for eal 3 students: Australia the movie

A Independência da América Espanhola LAPBOOK.pdf

A Independência da América Espanhola LAPBOOK.pdf

What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...

What is Digital Literacy? A guest blog from Andy McLaughlin, University of Ab...

- 1. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Profundizar y contextualizar el conocimiento de la Unidad 3. Paso 4 Marleny Parra Romero Nidia Mayerly Carvajal Rocha Samuel David Rojas Baquero Grupo: 29 Universidad Nacional Abierta y a Distancia (UNAD) 07 Mayo de 2021
- 2. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Cónicas Hipérbola y elipse ¿Qué es una hipérbola? La Hipérbola es un conjunto de puntos en el plano (x, y) cuya diferencia a dos puntos fijos llamados focos es constante. ¿Cuáles son sus parámetros? Centro: C (h, k). Equidistante a los vértices Vértices V y V’ Donde las curvas se dividen en dos partes iguales. Focos: F y F’: Los puntos fijos. Eje Transverso: Una recta que para por los vértices y por los focos. Eje Conjugado: En una recta perpendicular al eje transverso y para por el centro. Asíntotas: Dos rectas que paran por el centro delimitan las curvas de la hipérbola.
- 3. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA • Grafica parámetros de la hipérbola
- 4. CÓNICA EN FORMA CANÓNICA La ecuación canónica o segmentaria de la recta, es la expresión algebraica de la recta que se determina conociendo a los valores dónde la recta corta a cada uno de los ejes coordenados. El valor donde la recta corta al eje X le llamaremos a, y el valor donde la recta corta al eje Y le llamaremos b, generando los dos puntos en el plano cartesiano (a, 0) y (0, b) respectivamente. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 5. Ecuación canónica • a es la abscisa en el origen de la recta. • b es la ordenada en el origen de la recta. • El independiente de la general NO debe ser cero, significa que la forma canónica de la recta NO describe a las rectas que pasan por el origen, ya que ahí a=b=0 • Si A o B de la ecuación general son cero, significa que la recta es horizontal o vertical respectivamente, lo que lleva a que a o b de la ecuación canónica no existen, entonces tampoco hay forma de la ecuación canónica para este caso. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 6. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Ecuación canónica: (con eje mayor x) y ( eje mayor en y) Una Hipérbola con centro en (h, k) y eje transverso paralelo al eje x, tiene como ecuación: Una Hipérbola con centro en (h, k) y eje transverso paralelo al eje y, tiene como ecuación:
- 7. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Formulario de la hipérbola para resolver ejercicios
- 8. Ejemplo, resolución de ejercicios Hiperbólicos hipérbola con centro en el Origen ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 9. Hipérbola con centro fuera del Origen Determine los parámetros de una hipérbola que tiene como ecuación: Solución: De la ecuación: 𝑎2 = 9 ⟹⟹ 𝑎 = 3 y 𝑏2 = 4 ⟹⟹ 𝑏 = 2 Como el valor mayor esta sobre la variable x, el eje transverso esta sobre x. Par obtener “c”, podemos hacer uso de la siguiente condición: ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 10. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 11. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 12. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 13. ckaskdjds ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 14. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 15. Elipse ¿Qué es una Elipse? La elipse es un conjunto de puntos (x, y) en el plano cartesiano, tal que la suma de sus distancias a dos puntos fijos llamados focos, es constante. ¿Cuáles son sus parámetros? Centro: C (h, k) Vértices mayores: V y V’ Vértices menores: u y u’ Focos: f y f’ Eje mayor: 2a (Distancia V V ‘) Eje menor: 2b (Distancia u u ‘) ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 16. Ecuación canónica: (con eje mayor x) y ( eje mayor en y) Al igual que en la circunferencia, en la elipse la situación es similar. El centro es (h, k) que se obtiene cuando el centro que estaba en el origen se desplazo h unidades en x y k unidades en y, conlleva a ecuaciones canónicas ajustadas, las cuales son más generales. La ecuación canónica de una elipse con centro en (h, k) y eje mayor paralelo al eje x es: De la misma manera, la ecuación canónica de una elipse con centro en (h, k) y eje mayor paralelo al eje y es: ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 17. Ejemplo, resolución de ejercicios Elipse • Dada la elipse de ecuación: 𝑥−6 2 36 + 𝑦−4 2 16 = 1 , hallar su centro, semiejes, vértices y focos. Solución Dado que tenemos la ecuación en su forma canónica, tenemos que el centro es 𝐶 = 6; 4 Para hallar los semiejes, tenemos que: 𝑎2 = 36 ⇒⇒ 𝑎 = 6 y 𝑏2 = 16 ⇒⇒ 𝑏 = 4 Como el mayor valor esta sobre x entonces este sería el semieje mayor y (y) el semieje menor ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Ahora, notemos que dado que a representa el semieje mayor, y este divide a la expresión 𝑥 − 6 2 entonces el eje mayor de la elipse es paralelo al eje de las abscisas, esto indica que los vértices están a unidades a la derecha y a unidades a la izquierda del centro, así, los vértices son: 𝑉1 = 0; 4 y 𝑉2 = 12; 4 .
- 18. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA Por último, encontremos los focos. Tenemos que la mitad de la distancia focal (la distancia del centro de la elipse a cualquier de sus focos) se denota por y cumple que , dicho esto, tenemos que . Así, tenemos que y los focos son y . . Bosquejo de la gráfica
- 19. Ejemplo, resolución de ejercicios Elipse ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 20. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 21. Resolución de Problemas – Geometría Analítica “La geometría analítica es una fuerte herramienta matemática para resolver problemas de las diversas áreas del conocimiento”. (Rondón, J. 2017) Lineamientos a seguir: 1. Leer detenidamente el problema propuesto, para así comprender que nos está solicitando hallar. 2. Establecer que figura se adapta al problema a partir de los datos encontrados en el problema y los que se deben hallar. 3. Conforme a los datos obtenidos, identificar y aplicar la ecuación que se ajuste para dar solución problema. 4. Hallar la solución a las preguntas planteadas en el problema. A continuación un ejemplo de lo mencionado anteriormente: ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 22. 1. Leer detenidamente el problema: Nos dice que el planeta Mercurio se mueve en forma elíptica alrededor del sol con una excentricidad y su eje mayor. Así que debemos hallar la distancia máxima entre estos. 2. Establecer la figura que se adapta al problema: En el problema nos dice que es una forma elíptica. 3. Identificar y aplicar la ecuación que se ajuste para dar solución problema: Conforme a los datos de la excentricidad=0,206 y el eje mayor es de 0,774 Unidades Astronómicas. Siendo así, tenemos que: ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 23. 4. Hallar la solución a las preguntas planteadas en el problema. ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 24. Referencias Bibliográficas Rondón, J. (2017). Algebra, Trigonometría y Geometría Analítica. Bogotá D.C.: Universidad Nacional Abierta y a Distancia. Páginas 237 – 265. Recuperado de https://repository.unad.edu.co/handle/10596/11583 Ortiz Ceredo, F. J. Ortiz Ceredo, F. J. y Ortiz Ceredo, F. J. (2018). Matemáticas 3 (2a. ed.). Grupo Editorial Patria. https://elibro- net.bibliotecavirtual.unad.edu.co/es/ereader/unad/40539?page=51 Real, M. (2010). Secciones Cónicas. Recuperado de https://repository.unad.edu.co/handle/10596/7690 ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA
- 25. GRACIAS POR SU ATENCIÓN ALGEBRA, TRIGONOMETRÍA Y GEOMETRÍA ANALITICA