Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This document contains solutions to exercises on conic sections (hyperbolas, ellipses, circles, and parabolas) from a geometry guide.
The solutions include finding the center, vertices, foci, and eccentricity of various hyperbolas and ellipses given in standard form. One example given is a circle, for which the center and radius are identified. Another example is completed by rewriting the equation in canonical form.
The purpose is to understand these geometry topics for future professional careers by solving the guide's problems and verifying answers using GeoGebra.
1. This document contains 20 multiple part questions about differential equations. The questions cover topics like determining the degree and order of differential equations, solving differential equations, identifying whether equations are homogeneous, and forming differential equations to represent families of curves with given properties.
2. The questions range from 1 to 6 marks and include both conceptual questions about differential equations as well as problems requiring solving specific equations. A variety of solution techniques are required including separating variables, homogeneous property, and identifying particular solutions given initial conditions.
3. The document tests mastery of fundamental differential equation concepts and skills like classification, solving, identifying homogeneous property, and setting up equations to model geometric situations. A solid understanding of differential equations is needed to successfully answer all
The document provides bibliographic references for 14 books and papers on the topics of tensors, vector analysis, and continuum mechanics. It includes publication information such as author names, titles, publishers, and years. The references are listed alphabetically by author surname.
IIT JAM MATH 2018 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2018 Question Paper
IIT JAM Preparation Strategy
For full solutions contact us.
Call - 9836793076
IIT JAM MATH 2019 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Preparation Strategy
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2019 Question Paper
For full solutions contact us.
Call - 9836793076
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
This document contains solutions to problems from calculus and multivariable calculus courses. It begins with single variable calculus problems involving tangent lines, integrals, derivatives, and infinite series. The second part involves problems related to parametric equations, vectors, planes, cylinders, and graphing surfaces. The last part contains problems involving level curves, least squares regression, and using computer algebra systems to plot functions.
IIT JAM MATH 2020 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2020 Question Paper
IIT JAM Preparation Strategy
For full solutions contact us.
Call - 9836793076
This document contains solutions to exercises on conic sections (hyperbolas, ellipses, circles, and parabolas) from a geometry guide.
The solutions include finding the center, vertices, foci, and eccentricity of various hyperbolas and ellipses given in standard form. One example given is a circle, for which the center and radius are identified. Another example is completed by rewriting the equation in canonical form.
The purpose is to understand these geometry topics for future professional careers by solving the guide's problems and verifying answers using GeoGebra.
1. This document contains 20 multiple part questions about differential equations. The questions cover topics like determining the degree and order of differential equations, solving differential equations, identifying whether equations are homogeneous, and forming differential equations to represent families of curves with given properties.
2. The questions range from 1 to 6 marks and include both conceptual questions about differential equations as well as problems requiring solving specific equations. A variety of solution techniques are required including separating variables, homogeneous property, and identifying particular solutions given initial conditions.
3. The document tests mastery of fundamental differential equation concepts and skills like classification, solving, identifying homogeneous property, and setting up equations to model geometric situations. A solid understanding of differential equations is needed to successfully answer all
The document provides bibliographic references for 14 books and papers on the topics of tensors, vector analysis, and continuum mechanics. It includes publication information such as author names, titles, publishers, and years. The references are listed alphabetically by author surname.
IIT JAM MATH 2018 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2018 Question Paper
IIT JAM Preparation Strategy
For full solutions contact us.
Call - 9836793076
IIT JAM MATH 2019 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Preparation Strategy
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2019 Question Paper
For full solutions contact us.
Call - 9836793076
This document appears to be the table of contents and problems from Chapter 0 of a mathematics textbook. The table of contents lists 17 chapters and their corresponding page numbers. The problems cover a range of algebra topics including integers, rational numbers, properties of operations, solving equations, and rational expressions. There are over 70 problems presented without solutions for students to work through.
This document contains solutions to problems from calculus and multivariable calculus courses. It begins with single variable calculus problems involving tangent lines, integrals, derivatives, and infinite series. The second part involves problems related to parametric equations, vectors, planes, cylinders, and graphing surfaces. The last part contains problems involving level curves, least squares regression, and using computer algebra systems to plot functions.
IIT JAM MATH 2020 Question Paper | Sourav Sir's ClassesSOURAV DAS
IIT JAM Math Previous Year Question Paper
IIT JAM Math 2020 Question Paper
IIT JAM Preparation Strategy
For full solutions contact us.
Call - 9836793076
The document provides examples and explanations of finding equations of circles that are tangent to lines or pass through given points. It discusses using formulas for perpendicular distance from a line or point to determine a circle's radius and center. It also shows solving systems of equations algebraically or using geometric constructions to find centers and radii when given additional constraints like points or lines that a circle must be tangent to or pass through.
This document is a 50 question practice test for the SAT Mathematics Level 2 exam. It provides 1 hour (60 minutes) to complete the test. The test contains multiple choice questions covering a variety of math topics, including geometry, trigonometry, algebra, statistics, and other concepts. Answers are selected from 5 possible choices labeled A through E.
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
This document provides examples of solutions to problems involving number theory, algebra, geometry, and probability. It contains the following types of problems:
- Number theory word problems involving finding missing terms in sequences
- Solving quadratic equations and determining equations with transformed roots
- Calculating areas and perimeters of geometric shapes like octagons and hexagons
- Finding equations of lines transformed by parallel shifts
- Probability problems involving counting outcomes, finding probabilities of events, and calculating combinations and permutations
The document aims to demonstrate smart solutions to a variety of mathematical problems across different topics through clear explanations and step-by-step working.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
1. The document defines various functions and relations using set-builder and function notation.
2. Examples of linear, quadratic, and polynomial functions are provided with their domain and range restrictions.
3. Common transformations of basic quadratic functions like y=x^2 are demonstrated, such as shifting the graph left or right and changing the sign of coefficients.
The document describes the midpoint circle algorithm for drawing circles on a pixel screen. It explains how the algorithm determines the midpoint between the next two possible consecutive pixels and checks if the midpoint is inside or outside the circle to determine which pixel to illuminate. It provides the mathematical equations and steps used to iteratively calculate the x and y coordinates of each pixel on the circle. The algorithm is implemented in a C++ program to draw a circle on a graphics screen.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
This document provides an overview of key concepts in differential equations including:
- A differential equation contains derivatives of dependent variables with respect to independent variables.
- The order of a differential equation is defined as the order of the highest derivative. The degree is the highest power of the highest order derivative.
- Differential equations can be formed by differentiating curves and eliminating arbitrary constants.
- Common methods for solving differential equations include variable separation, homogeneous equations, and linear equations.
This document contains a chapter on functions with 30 math exercises. The exercises involve evaluating functions, determining domains and ranges, analyzing graphs of functions, and solving word problems involving functions.
This section introduces differential equations and their use in mathematical modeling. It provides examples of verifying solutions to differential equations by direct substitution. Typical problems show finding an integrating constant to satisfy an initial condition. Differential equations are derived from descriptions of real-world phenomena involving rates of change. The section establishes foundational knowledge of differential equations and their solution methods.
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System dynamics 3rd edition palm solutions manual
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This document contains the mark scheme for a mathematics exam involving several multi-part questions.
In question 1, students could earn up to 3 marks for correctly factorizing a quadratic expression in one or two steps.
Question 2 was worth up to 2 marks for correctly writing the equation of a straight line in y=mx+c form.
Question 3 involved solving equations and inequalities across three parts, with a total of 6 available marks through setting up and solving the relevant expressions.
The remaining questions addressed topics including arithmetic and geometric sequences, calculus, coordinate geometry, and quadratic functions. Students could earn marks for setting up correct expressions and equations and obtaining the right numerical or algebraic solutions at each stage.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
This document contains exercises involving functions and graphs. There are multiple choice and free response questions testing understanding of properties of functions including domain, range, intercepts, maxima/minima, and graphing functions. Solutions are provided for checking work.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
The document provides practice questions on surds and indices, differentiation, integration, and quadratics. For surds and indices, questions involve simplifying expressions with surds and rationalizing denominators. Differentiation questions involve taking derivatives of functions and finding derivatives from given information. Integration questions involve taking integrals and finding functions from their derivatives. Quadratic questions involve solving quadratic equations, finding the discriminant, and determining conditions on constants for the equation to have certain root properties.
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.
The document discusses different types of conic sections including parabolas, circles, ellipses, and hyperbolas. It provides the standard formulas for each type of conic section and explains how to find the x-intercepts and domain by substituting values into the equation or analyzing the graph. Key information covered includes the standard forms of parabolas as y=ax^2+bx+c or x=ay^2+by+c, circles as (x-h)^2+(y-k)^2=r^2, ellipses as x^2/a^2 + y^2/b^2 = 1 or x^2/b^2 + y^2/a^
“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.
The document provides examples and explanations of finding equations of circles that are tangent to lines or pass through given points. It discusses using formulas for perpendicular distance from a line or point to determine a circle's radius and center. It also shows solving systems of equations algebraically or using geometric constructions to find centers and radii when given additional constraints like points or lines that a circle must be tangent to or pass through.
This document is a 50 question practice test for the SAT Mathematics Level 2 exam. It provides 1 hour (60 minutes) to complete the test. The test contains multiple choice questions covering a variety of math topics, including geometry, trigonometry, algebra, statistics, and other concepts. Answers are selected from 5 possible choices labeled A through E.
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
This document provides examples of solutions to problems involving number theory, algebra, geometry, and probability. It contains the following types of problems:
- Number theory word problems involving finding missing terms in sequences
- Solving quadratic equations and determining equations with transformed roots
- Calculating areas and perimeters of geometric shapes like octagons and hexagons
- Finding equations of lines transformed by parallel shifts
- Probability problems involving counting outcomes, finding probabilities of events, and calculating combinations and permutations
The document aims to demonstrate smart solutions to a variety of mathematical problems across different topics through clear explanations and step-by-step working.
This document provides examples of transformations involving complex variables and their applications. It contains 3 examples of inversion transformations where a line or circle in the z-plane is transformed to a circle or line in the w-plane. It also contains 2 examples of square transformations where a region in the z-plane is transformed to parabolic regions in the w-plane. Additionally, it discusses finding the image of a line or circle under translations in the complex plane.
1. The document defines various functions and relations using set-builder and function notation.
2. Examples of linear, quadratic, and polynomial functions are provided with their domain and range restrictions.
3. Common transformations of basic quadratic functions like y=x^2 are demonstrated, such as shifting the graph left or right and changing the sign of coefficients.
The document describes the midpoint circle algorithm for drawing circles on a pixel screen. It explains how the algorithm determines the midpoint between the next two possible consecutive pixels and checks if the midpoint is inside or outside the circle to determine which pixel to illuminate. It provides the mathematical equations and steps used to iteratively calculate the x and y coordinates of each pixel on the circle. The algorithm is implemented in a C++ program to draw a circle on a graphics screen.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and finding the nearest integer of a difference. For algebra, it solves systems of equations and determines values based on given equations. For geometry, it calculates areas and volumes. For probability, it finds probabilities of arrangements and outcomes of dice rolls and ball draws.
1. The document presents problems involving number theory, algebra, geometry, and probability. For number theory, it provides exercises and solutions involving sums of powers and nearest integers. For algebra, it solves systems of equations and determines values based on relationships between variables. For geometry, it calculates areas and volumes. For probability, it determines probabilities of events occurring based on arrangements and selections from sets.
This document provides an overview of key concepts in differential equations including:
- A differential equation contains derivatives of dependent variables with respect to independent variables.
- The order of a differential equation is defined as the order of the highest derivative. The degree is the highest power of the highest order derivative.
- Differential equations can be formed by differentiating curves and eliminating arbitrary constants.
- Common methods for solving differential equations include variable separation, homogeneous equations, and linear equations.
This document contains a chapter on functions with 30 math exercises. The exercises involve evaluating functions, determining domains and ranges, analyzing graphs of functions, and solving word problems involving functions.
This section introduces differential equations and their use in mathematical modeling. It provides examples of verifying solutions to differential equations by direct substitution. Typical problems show finding an integrating constant to satisfy an initial condition. Differential equations are derived from descriptions of real-world phenomena involving rates of change. The section establishes foundational knowledge of differential equations and their solution methods.
System dynamics 3rd edition palm solutions manualSextonMales
System dynamics 3rd edition palm solutions manual
Full download: https://goo.gl/7Z6QZ3
People also search:
system dynamics palm 3rd edition pdf
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system dynamics palm 3rd edition free pdf
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system dynamics 3rd edition ogata pdf
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This document contains the mark scheme for a mathematics exam involving several multi-part questions.
In question 1, students could earn up to 3 marks for correctly factorizing a quadratic expression in one or two steps.
Question 2 was worth up to 2 marks for correctly writing the equation of a straight line in y=mx+c form.
Question 3 involved solving equations and inequalities across three parts, with a total of 6 available marks through setting up and solving the relevant expressions.
The remaining questions addressed topics including arithmetic and geometric sequences, calculus, coordinate geometry, and quadratic functions. Students could earn marks for setting up correct expressions and equations and obtaining the right numerical or algebraic solutions at each stage.
The document discusses solving the 8 queens problem using backtracking. It begins by explaining backtracking as an algorithm that builds partial candidates for solutions incrementally and abandons any partial candidate that cannot be completed to a valid solution. It then provides more details on the 8 queens problem itself - the goal is to place 8 queens on a chessboard so that no two queens attack each other. Backtracking is well-suited for solving this problem by attempting to place queens one by one and backtracking when an invalid placement is found.
This document contains exercises involving functions and graphs. There are multiple choice and free response questions testing understanding of properties of functions including domain, range, intercepts, maxima/minima, and graphing functions. Solutions are provided for checking work.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
The document provides practice questions on surds and indices, differentiation, integration, and quadratics. For surds and indices, questions involve simplifying expressions with surds and rationalizing denominators. Differentiation questions involve taking derivatives of functions and finding derivatives from given information. Integration questions involve taking integrals and finding functions from their derivatives. Quadratic questions involve solving quadratic equations, finding the discriminant, and determining conditions on constants for the equation to have certain root properties.
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.
The document discusses different types of conic sections including parabolas, circles, ellipses, and hyperbolas. It provides the standard formulas for each type of conic section and explains how to find the x-intercepts and domain by substituting values into the equation or analyzing the graph. Key information covered includes the standard forms of parabolas as y=ax^2+bx+c or x=ay^2+by+c, circles as (x-h)^2+(y-k)^2=r^2, ellipses as x^2/a^2 + y^2/b^2 = 1 or x^2/b^2 + y^2/a^
“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.
This document discusses applications of various conic sections in real life. It begins by defining conic sections as curves derived from slicing a double-napped cone and lists the main types - parabolas, ellipses, circles, and hyperbolas. It then provides examples of applications for each type of conic section, such as parabolas in football trajectories, ellipses in eye shapes and planet orbits, circles in wheels and records, and hyperbolas in sonic booms and lighthouse beams. The document aims to illustrate how conic sections appear frequently in architecture, engineering, and natural phenomena.
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This document defines and provides examples of different types of conic sections - parabolas, ellipses, and hyperbolas. It explains that conic sections are curves formed by the intersection of a plane with a cone, and that points on a conic section have a fixed ratio between their distance to a focus point and its directrix line, known as eccentricity. Eccentricity values distinguish the different conic section types. Examples of each in diagrams and applications like planetary orbits, bicycle gears, and network synchronization are also provided.
The document discusses equations of circles in various forms. It provides the general equation of a circle, as well as equations for circles given specific properties like center point and radius, diameter endpoints, tangency to an axis, and passing through a given point. Examples are worked through to find the equation of a circle matching given conditions or to obtain properties of a circle from its equation. Circles can be represented using forms based on the center and radius, diameter endpoints, or general equation.
Here are the steps to solve this problem:
1) The given equation is: y = x2 + 2x
2) To complete the table, we need to calculate the value of y when x = -3 and when x = 1
3) When x = -3:
y = (-3)2 + 2(-3)
y = 9 - 6
y = 3
4) When x = 1:
y = 12 + 2(1)
y = 1 + 2
y = 3
So the completed table is:
Table 1
x -3 1
y 3 3
(b) Sketch the graph of y = x2 + 2
Circles are geometric shapes defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Circle equations can be written in standard form (x-h)2 + (y-k)2 = r2, where (h, k) are the coordinates of the center and r is the radius. Problems involve finding the equation of a circle given its center and radius, graphing circles, rewriting equations in standard form, determining if an equation represents a circle, point or null set, finding equations of circles tangent to lines or passing through points, and finding equations of circles inscribed in or
This document provides a study package on circles for a mathematics class. It begins with an index listing the topics covered, which include theory, revision, exercises, assertion and reason questions, and past examination questions. It then covers circle theory, equations of circles in various forms including parametric and Cartesian, intercepts made by circles on axes, the position of points with respect to circles, lines and circles, and tangents to circles. Examples are provided to illustrate each concept. The document is intended to be a comprehensive resource for students to learn about circles.
This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.
1) The document provides information on circle equations and properties including: the general form of a circle equation, finding the center and radius from an equation, and determining if a point lies inside, outside or on a circle.
2) Examples are given for writing circle equations in different forms, finding centers and radii, and finding intersection points between circles and lines.
3) The key steps for finding intersections between a line and circle are outlined using simultaneous equations and the discriminant.
This document discusses straight line graphs and various methods for plotting straight lines on a graph. It covers identifying horizontal, vertical and diagonal lines, writing equations in y=mx+c form, using a table method to plot lines by substituting x-values, and using the x=0, y=0 method to find two points and plot a line from its equation. Exercises are provided for students to practice identifying and plotting different types of straight lines.
Circle, Definition, Equation of circle whose center and radius is known, General equation of a circle, Equation of circle passing through three given points, Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2), Tangent and Normal to a given circle at given point.
This document contains a request to provide a song or video link to play while waiting for classmates, as well as a request for names and adjectives starting with the same letter. It also contains information about circles, including definitions, properties, equations, examples of finding standard and general equations of circles given properties, and an example of finding the center and radius from a general equation.
The document describes several algorithms for drawing circles:
1. Using the circle equation requires significant computation and results in a poor appearance.
2. Using trigonometric functions is time-consuming due to trig computations.
3. The midpoint circle algorithm uses the midpoint between candidate pixels to determine which is closer to the actual circle. It has less computation than the circle equation.
4. Bresenham's circle algorithm uses a decision parameter D to iteratively select the next pixel, requiring fewer computations than trigonometric functions.
The document discusses Gauss Divergence Theorem and provides two examples of using it to evaluate surface integrals. Gauss Divergence Theorem states that the surface integral of a vector field F over a closed surface S enclosing a volume V is equal to the volume integral of the divergence of F over V. The first example uses this to evaluate a surface integral over a cylinder. The second example verifies Gauss Divergence Theorem for a vector field F over the surface of a cube.
The document provides information about the standard equation of a circle given its center and tangency condition. It gives the equation (x – 5)2 + (y + 6)2 = 36 for the circle with center (5,-6) tangent to the x-axis. It then explains how to rewrite this in general form as x2 + y2 - 10x +12y+25 =0. It also gives the general form of the equation of a circle Ax2 + By2 + Cx + Dy + E = 0 and examples of identifying the center and radius from equations in general form.
The document presents several problems involving finding equations of circles given properties such as the center and radius, points the circle passes through, tangency to lines, etc. There are over 20 subproblems across 4 sections - A focuses on finding center and radius of circles given equations, B focuses on finding equations of circles given properties like center and radius or points, C involves circles tangent to given lines, and D analyzes if equations represent circles, points or empty sets and finds properties if they are circles.
The document discusses the standard form of the equation of a circle. It can be written as x2 + y2 = r2 if the center is at the origin (0,0), or (x - h)2 + (y - k)2 = r2 if the center is at a point (h,k). Several examples are provided of writing the equation of a circle given its center and radius. The document also discusses determining the radius if the circle is tangent to the x-axis, y-axis, or a line.
This document provides an outline of topics in algebra including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples and explanations for each topic.
This document discusses how to graph and solve quadratic inequalities. It provides steps for graphing quadratic inequalities by sketching the parabola and shading the appropriate region based on a test point. Examples are given of solving quadratic inequalities graphically by determining the portions of the graph above or below the x-axis and obtaining the solution intervals. Exercises are also worked through to practice solving quadratic inequalities graphically.
This document provides a review of various algebra topics including: indices, expanding single and double brackets, substitution, solving equations, solving equations from angle problems, finding the nth term of sequences, simultaneous equations, inequalities, factorizing using common factors, quadratics, grouping and the difference of two squares. It also includes examples to practice each topic.
This document provides an overview of topics covered in intermediate algebra revision including: collecting like terms, multiplying terms, indices, expanding single and double brackets, substitution, solving equations, finding nth terms of sequences, simultaneous equations, inequalities, factorizing common factors and quadratics, solving quadratic equations, rearranging formulas, and graphing curves and lines. The document contains examples and practice problems for each topic.
Similar to 1511 circle position-conics project (20)
➽=ALL False flag-War Machine-War profiteering-Energy (oil/Gas) Iraq, Iran,…oil and gas
USA invades other countries just to own their natural resources and to place them in the hands of American corporations. Facebook doesn’t call that terrorism. They call it democracy. BBC, CNN, FOX NEWS, FR 24, ITV/CH 4, SKY, EURO NEWS, ITV trash Sun paper,… Facebook all are protector and preserver of the propaganda classifying IR Iran as a dangerous terrorist organization. But FB, BBC, CNN, FOX NEWS, FR 24, ITV/CH 4-SKY, EURO NEWS, ITV do know well, that USA is the biggest terrorist country in the world.
‘terrorism’ the unlawful use of violence and intimidation, especially against civilians, in the pursuit of political aims.
"the fight against terrorism" is the fight against the unlawful use of violence and intimidation and carpet bombing.
Ever since the beginning of the 19th century, the West has been sucking on the jugular vein of the Moslem body politic like a veritable vampire whose thirst for Moslem blood is never sated and who refused to let go. Since 1979, Iran, which has always played the role of the intellectual leader of the Islamic world, has risen up to put a stop to this outrage against God’s law and will, and against all decency.
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The document discusses construction productivity in the UK and other countries. It provides factors that impact productivity such as site/project management, resource management, labor characteristics, and motivation. Productivity in the UK construction industry has improved over the past decade but still lags countries like Germany and France. Reasons given for relatively lower UK productivity include issues with subcontracting, materials handling, training, and technology adoption. Improving areas like planning, prefabrication, and reducing waste could further increase construction productivity.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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 ..
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তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
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.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
1511 circle position-conics project
1. Introduction
A Closer Look at Graphing
Translations
Definition of a Circle
Definition of Radius
Definition of a Unit Circle
Deriving the Equation of a Circle
The Distance Formula
Moving The Center
Completing the Square
It is a Circle if?
Conic Movie
Conic Collage
End Next
Geometers Sketchpad: Cosmos
Geometers Sketchpad: Headlights
Projectile Animation
Planet Animation
Plane Intersecting a Cone Animation
Footnotes
Skip Intro
3. The quadratic relations that we studied in the beginning
of the year were in the form of y = Axy = Ax22
+ Dx + F+ Dx + F,
where A, D, and F stand for constants, and A ≠ 0.
This quadratic relation is a parabola, and it is the
only one that can be a function.
It does not have to be a function, though.
A parabola is determined by a plane intersecting a
cone and is therefore considered a conic section.
NextPreviousMain MenuEnd
4. The general equation for all conic sections is:
AxAx22
+ Bxy + Cy+ Bxy + Cy22
+ Dx + Ey + F = 0+ Dx + Ey + F = 0
where A, B, C, D, E and F represent constants
When an equation has a “yy22
” term and/or an “xyxy”
term it is a quadratic relation instead of a quadratic
function.
and where the equal sign could be replaced by an
inequality sign.
NextPreviousMain MenuEnd
5. A Closer Look at Graphing Conics
Plot the graph of each relation. Select values of x and
calculate the corresponding values of y until there are
enough points to draw a smooth curve. Approximate all
radicals to the nearest tenth.
1)1) xx22
+ y+ y22
= 25= 25
2)2) xx22
+ y+ y22
+ 6x = 16+ 6x = 16
3)3) xx22
+ y+ y22
- 4y = 21- 4y = 21
4)4) xx22
+ y+ y22
+ 6x - 4y = 12+ 6x - 4y = 12
5)5) ConclusionsConclusions
NextPreviousMain MenuLast ViewedEnd
6. xx22
+ y+ y22
= 25= 25
x = 3
32
+ y2
= 25
9 + y2
= 25
y2
= 16
y2
=16
y =4
There are 22 points to graph: (3,4)
(3,-4)
y = ± 4
NextPreviousMain MenuA Close Look at GraphingEnd
7. Continue to solve in this manner, generating a table of values.
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0
NextPreviousMain MenuA Close Look at GraphingEnd
8. Graphing xx22
+ y+ y22
= 25= 25
x y
-5 0
-4 ±3
-3 ±4
-2 ±4.6
-1 ±4.9
0 ±5
1 ±4.9
2 ±4.6
3 ±4
4 ±3
5 0
x
y
-10 -5 5
6
4
2
-2
-4
-6
NextPreviousMain MenuA Close Look at GraphingEnd
9. xx22
+ y+ y22
+ 6x = 16+ 6x = 16
x = 1
12
+ y2
+6(1) = 16
1 + y2
+6 = 16
y2
= 9
y2
=9
y =3
There are 22 points to graph: (1,3)
(1,-3)
y = ± 3
NextPreviousMain MenuA Close Look at GraphingEnd
10. Continue to solve in this manner, generating a table of values.
x y
-8 0
-7 ±3
-6 ±4
-5 ±4.6
-4 ±4.9
-3 ±5
-2 ±4.9
-1 ±4.6
0 ±4
1 ±3
2 0
x y
-8 0
-7 ±3
-6 ±4
-5 ±4.6
-4 ±4.9
-3 ±5
-2 ±4.9
-1 ±4.6
0 ±4
1 ±3
2 0
NextPreviousMain MenuA Close Look at GraphingEnd
18. -10 -5 5 10
8
6
4
2
-2
-4
xx22
+ y+ y22
= 25= 25
xx22
+ y+ y22
+ 4x = 16+ 4x = 16
xx22
+ y+ y 22
- 6y = 21- 6y = 21
xx22
+ y+ y22
+ 4x - 6y = 12+ 4x - 6y = 12
What conclusions can you draw about the shape and location
of the graphs?
NextPreviousMain MenuA Close Look at GraphingEnd
19. What conclusions can you draw about the shape and location
of the graphs?
• All of the graphs are circles. xx22
+ y+ y22
= 25= 25
10
5
-5
-10 10
• As you add an x-term, the graph moves left. xx22
+ y+ y22
+ 4x = 16+ 4x = 16
• As you subtract a y-term, the graph moves up. xx22
+ y+ y 22
- 6y = 21- 6y = 21
• You can move both left and up. xx22
+ y+ y22
+ 4x - 6y = 12+ 4x - 6y = 12
NextPreviousMain MenuA Close Look at GraphingEnd
20. A circlecircle is the set of all points in a plane equidistant from a
fixed point called the centercenter.
-10 -5 5 10
6
4
2
-2
-4
-6
Center
CircleCircle
Center
GEOMETRICAL DEFINITION OF A
CIRCLE
NextPreviousMain MenuEnd
21. A radiusradius is the segment whose endpoints are the center of
the circle, and any point on the circle.
6
4
2
-2
-4
-6
-10 -5 5 10
radiusradius
radius
radius
NextPreviousMain MenuEnd
22. A unit circle is a circle with a radius of 1 whose center is at
the origin.
It is the circle on which all other circles are based.
-2 -1 1 2
1.5
1
0.5
-0.5
-1
A unit circle is a circle with a radius of 1 whose center is at
the origin.
r =
1
r =
1
A unit circle is a circle with a radius of 1 whose center is at
the origin.
Center: (0, 0)Center: (0, 0)
A unit circleunit circle is a circle with a radius of 1radius of 1 whose center is atcenter is at
the originthe origin.
NextPreviousMain MenuEnd
23. The equation of a circle is derived from its radiusradius.
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
NextPreviousMain MenuEnd Last Viewed
24. Use the distance formuladistance formula to find an equation for x and y.
This equation is also the equation for the circlethe equation for the circle.
NextPreviousMain Menu
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
Deriving the Equation of a CircleEnd
25. THE DISTANCE FORMULA
(x2, y2)
(x1, y1)
D =(x2 −x1)2
+(y2 −y1)2
NextPreviousMain Menu
1
0.5
-0.5
-1
-1.5
-2 -1 1 2
Deriving the Equation of a CircleEnd
26. Deriving the Equation of a Circle
D =x2 −x1( )
2
+y2 −y1( )
2
r =x −0( )
2
+y −0( )
2
r =x2
+y2
r2
= x2
+y2
( )
2
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
Let rr for radiusradius length replace D for distance.
r2
= x2
+ y2
rr22
= x= x22
+ y+ y22
Is the equation for a circle
with its center at the origin
and a radius of length r.
NextPreviousMain MenuEnd
27. The unit circle therefore has the equation:
x2
+ y2
= 1
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
r=
1
r=
1
NextPreviousMain MenuEnd Deriving the Equation of a Circle
28. x2
+ y2
= 4
If r = 2r = 2, then
1
0.5
-0.5
-1
-1.5
-2 -1 1
(x, y)(x, y)
(0, 0)(0, 0)
r=
2
r=
2
NextPreviousMain MenuDeriving the Equation of a CircleEnd
29. In order for a satellite to remain in a circular orbit above the
Earth, the satellite must be 35,000 km above the Earth.
Write an equation for the orbit of the satellite. Use the
center of the Earth as the origin and 6400 km for the radius
of the earth.
(x - 0)2
+ (y - 0)2
= (35000 + 6400)2
x2
+ y2
= 1,713,960,000
NextPreviousMain MenuDeriving the Equation of a CircleEnd
30. What will happen to the equation if the center is not at the
origin?
6
4
2
-2
-4
-6
-10 -5 5
r =x2 −x1( )
2
+y2 −y1( )
2
(1,2)(1,2)
r = 3
r = 3 (x,y)(x,y)
3 =x −1()
2
+y −2()
2
32
= x −1( )
2
+y −2( )
2
2
9 =x −1()
2
+y −2( )
2
NextPreviousMain MenuEnd Last Viewed
31. -10 -5 5 10
No matter where the circle is located, or where the center
is, the equation is the same.
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
(h,k)(h,k)
(x,y)(x,y)
rr
NextPreviousMain MenuMoving the CenterEnd
32. 6
4
2
-2
-4
-10 -5 5 10
(x,y)(x,y)
Assume (x, y)(x, y) are the coordinates of a point on a circle.
(h,k)(h,k)
The center of the circle is (h, k)(h, k),
rr
and the radius is r.
Then the equation of a circle is: ((xx -- hh))22
+ (+ (yy -- kk))22
== rr22
.
NextPreviousMain MenuMoving the CenterEnd
33. ((xx -- hh))22
+ (+ (yy -- kk))22
== rr22
((xx -- 00))22
+ (+ (yy -- 33))22
== 7722
(x)(x)22
+ (y - 3)+ (y - 3)22
= 49= 49
Write the equation of a circle with a center at (0, 3)center at (0, 3) and a
radius of 7radius of 7.
20
15
10
5
-5
-20 -10 10 20
NextPreviousMain MenuMoving the CenterEnd
34. Find the equation whose diameter has endpoints of
(-5, 2) and (3, 6).
First find the midpoint of the diameter using the
midpoint formula.
MIDPOINT
M(x,y) =
x1 +x2
2
,
y1 +y2
2
This will be the center.
M(x,y) =
−5 +3
2
,
2 +6
2
M(x,y) =−1, 4( )
NextPreviousMain MenuMoving the CenterEnd
35. Find the equation whose diameter has endpoints of
(-5, 2) and (3, 6).
Then find the length distance between the midpoint
and one of the endpoints.
DISTANCE FORMULA
This will be the radius.
D =x2 −x1( )
2
+y2 −y1( )
2
D =−1−3( )
2
+4 −6()
2
D =−4()
2
+−2()
2
D =16+4
D =20
NextPreviousMain MenuMoving the CenterEnd
36. Find the equation whose diameter has endpoints of
(-5, 2) and (3, 6).
Therefore the center is (-1, 4)
The radius is 20
(x - -1)2
+ (y - 4)2
= 20
2
(x + 1)2
+ (y - 4)2
= 20
-10
10
5
-5
NextPreviousMain MenuMoving the CenterSkip TangentsEnd
37. A line in the plane of a circle can intersectintersect the circle in 11 or
22 points. A line that intersects the circle in exactlyexactly oneone
point is said to be tangenttangent to the circle. The line and the
circle are considered tangent to each other at this point of
intersection.
Write an equation for a circle with center (-4, -3) that is
tangent to the x-axis.
-5
-2
-4
-6
A diagramdiagram will help.
(-4, 0)(-4, 0)
(-4, -3)(-4, -3)
33
A radius is always perpendicularperpendicular
to the tangent line.
(x + 4)2
+ (y + 3)2
= 9
NextPreviousMain MenuMoving the CenterEnd
38. The standard form equation for all conic sections is:
AxAx22
+ Bxy + Cy+ Bxy + Cy22
+ Dx + Ey + F = 0+ Dx + Ey + F = 0
where A, B, C, D, E and F represent constants and where the
equal sign could be replaced by an inequality sign.
How do you put a standard form equation into graphing form?
The transformation is accomplished through completing thecompleting the
squaresquare.
NextPreviousMain MenuEnd Last Viewed
39. Graph the relation x2
+ y2
- 10x + 4y + 13 = 0.13
1. Move the FF term to the other side.
x2
+ y2
- 10x + 4y = -13
3. Complete the square for the x-termsx-terms and y-termsy-terms.
xx22
-- 10x10x ++ yy22
++ 4y4y = -13= -13
−10
2
=−5
−5()
2
=25
4
2
= 2
2()
2
=4
xx22
-- 10x + 2510x + 25 ++ yy22
++ 4y + 44y + 4 = -13= -13 + 25+ 25 + 4+ 4
(x - 5)(x - 5)22
++ (y + 2)(y + 2)22
== 1616
xx22
++ yy22
-- 10x10x ++ 4y4y ++ 1313 = 0= 0
xx22
++ yy22
-- 10x10x ++ 4y4y == -13-13
2. Group the x-termsx-terms and y-termsy-terms together
xx22
-- 10x10x ++ yy22
++ 4y4y == -13-13
Graph the relation
NextPreviousMain MenuCompleting the SquareEnd
41. What if the relation is an inequality?
xx22
+ y+ y22
- 10x + 4y + 13 < 0- 10x + 4y + 13 < 0
Do the same steps to transform it to graphing form.
(x - 5)(x - 5)22
+ (y + 2)+ (y + 2)22
< 4< 422
This means the values are inside the circle.
The values are less thanless than the radiusradius.
NextPreviousMain MenuCompleting the SquareEnd
-5 5 10
6
4
2
-2
-4
-6
42. Write xx22
+ y+ y22
+ 6x - 2y - 54 = 0+ 6x - 2y - 54 = 0 in graphing form. Then
describe the transformation that can be applied to the
graph of x2
+ y2
= 64 to obtain the graph of the given
equation.
1. x2
+ y2
+ 6x - 2y = 54
2. x2
+ 6x + y2
- 2y = 54
3. (6
/2) = 3 (-2
/2) = -1
4. (3)2
= 9 (-1)2
= 1
5. x2
+ 6x + 9 + y2
- 2y + 1 = 54 + 9 + 1
6. (x + 3)2
+ (y - 1)2
= 64
7. (x + 3)2
+ (y - 1)2
= 82
8. center: (-3, 1) radius = 8
NextPreviousMain MenuCompleting the SquareEnd
43. Write xx22
+ y+ y22
+ 6x - 2y - 54 = 0+ 6x - 2y - 54 = 0 in graphing form. Then
describe the transformation that can be applied to the
graph of xx22
+ y+ y22
= 64= 64 to obtain the graph of the given
equation.
xx22
+ y+ y22
= 64= 64 is translated 3 units left and one
unit up to become (x + 3)(x + 3)22
+ (y - 1)+ (y - 1)22
= 64= 64.
10
5
-5
-10
-20 -10 10 20
NextPreviousMain MenuCompleting the SquareEnd
44. The graph of a quadratic relation will be a circlecircle if
the coefficientscoefficients of the xx22
term and yy22
term are
equalequal (and the xy term is zero).
PreviousMain MenuEnd