This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides 30 problems involving vectors. The problems cover topics such as determining whether vectors are collinear or orthogonal, calculating angles between vectors, finding vector equations of lines, and solving geometry problems using vectors. Solutions are provided for each problem.
The document discusses key concepts for straight line equations in GCSE mathematics including:
- Understanding that equations of the form y=mx+c correspond to straight line graphs
- Plotting graphs from their equations and finding gradients and intercepts
- Relating gradients to parallel and perpendicular lines
- Generating equations for lines parallel or perpendicular to given lines
- Finding gradients and equations from two points on a line
This document provides an introduction to direction cosines and direction ratios of lines in three-dimensional space, as well as equations of lines in space. It defines direction angles and explains how to calculate direction cosines from these angles. It also defines direction ratios and shows how they relate to direction cosines. The document gives examples of finding direction cosines and checking if points are collinear using direction ratios. It then derives equations for lines passing through a point and parallel to a vector, and lines passing through two given points, in both vector and Cartesian forms. Examples are provided to illustrate converting between forms.
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. Specifically, it asks the reader to: 1) Find direction ratios and determine perpendicularity between lines; 2) Write equations of planes passing through given points; and 3) Calculate lengths and points of intersection between lines and planes.
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. It covers topics such as direction cosines, perpendicular lines and planes, intersections of lines and planes, and vector and Cartesian representations of geometric entities in three dimensions.
This document defines key concepts in analytical geometry of three dimensions including planes, lines, angles between planes and lines, and conditions for coplanarity. It presents definitions for planes as loci of points satisfying linear equations, lines as ratios of distances from points, and direction ratios. Theorems provide equations for planes through points with given normals, angles between planes as angles between normals, lines through two points, conditions for coplanarity of lines, and angles between lines and planes. Skew (non-coplanar) lines are also defined.
This document contains information about straight lines and their various forms of equations. It discusses the slope-intercept form, intercept form, two-point form and general form of a straight line. It provides examples of finding the equation of a line given certain conditions like two points, slope and intercept, perpendicular or parallel lines etc. There are also examples of problems involving finding slope, intercepts, perpendicular or parallel lines to a given line. The document contains the necessary formulas and steps to solve such problems.
This document provides 30 problems involving vectors. The problems cover topics such as determining whether vectors are collinear or orthogonal, calculating angles between vectors, finding vector equations of lines, and solving geometry problems using vectors. Solutions are provided for each problem.
The document discusses key concepts for straight line equations in GCSE mathematics including:
- Understanding that equations of the form y=mx+c correspond to straight line graphs
- Plotting graphs from their equations and finding gradients and intercepts
- Relating gradients to parallel and perpendicular lines
- Generating equations for lines parallel or perpendicular to given lines
- Finding gradients and equations from two points on a line
This document provides an introduction to direction cosines and direction ratios of lines in three-dimensional space, as well as equations of lines in space. It defines direction angles and explains how to calculate direction cosines from these angles. It also defines direction ratios and shows how they relate to direction cosines. The document gives examples of finding direction cosines and checking if points are collinear using direction ratios. It then derives equations for lines passing through a point and parallel to a vector, and lines passing through two given points, in both vector and Cartesian forms. Examples are provided to illustrate converting between forms.
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. Specifically, it asks the reader to: 1) Find direction ratios and determine perpendicularity between lines; 2) Write equations of planes passing through given points; and 3) Calculate lengths and points of intersection between lines and planes.
This document contains 22 multi-part questions regarding three-dimensional geometry concepts such as finding direction ratios of lines, determining whether lines are perpendicular or parallel, finding equations of planes, and calculating distances between lines and points. It covers topics such as direction cosines, perpendicular lines and planes, intersections of lines and planes, and vector and Cartesian representations of geometric entities in three dimensions.
This document defines key concepts in analytical geometry of three dimensions including planes, lines, angles between planes and lines, and conditions for coplanarity. It presents definitions for planes as loci of points satisfying linear equations, lines as ratios of distances from points, and direction ratios. Theorems provide equations for planes through points with given normals, angles between planes as angles between normals, lines through two points, conditions for coplanarity of lines, and angles between lines and planes. Skew (non-coplanar) lines are also defined.
The document discusses lines and planes in 3D space. It defines lines as being determined by a point and direction vector, and gives parametric and symmetric equations to represent lines. Planes are defined by a point and normal vector, with standard and general forms for their equations. Methods are provided for finding the intersection of lines or planes, as well as the distance between a point and plane or line. Examples demonstrate finding equations of lines and planes, sketching planes, and determining relationships between lines or planes.
Straight Lines for JEE REVISION Power point presentationiNLUVWDu
The document provides information about the weightage and number of problems on straight lines that have been asked in JEE Main from 2020-2024. It lists important subtopics of straight lines like locus of a point, reflection, pair of lines, etc. It also provides various important formulas related to straight lines, short notes, and sample problems related to important subtopics. The document is a comprehensive reference material for the straight lines chapter of JEE Main exam preparation.
Assignment of straight lines and conic sectionKarunaGupta1982
1. The document contains 35 problems related to straight lines and conic sections like circles, ellipses, parabolas and hyperbolas. The problems involve finding equations of lines and conic sections given certain properties, finding properties like foci, vertices, axes, etc. from given equations.
2. Specific problems include finding slopes of lines, angle between lines, perpendicular lines, finding equations of lines passing through points or perpendicular/parallel to other lines.
3. For conic sections, problems include finding equations given foci, directrix, passing points, intersecting lines, centers and radii of circles, lengths of axes, foci, latus rectum and eccentricities of ellipses and hyperbol
This document discusses coordinates in space and three-dimensional coordinate geometry. It introduces points, lines, and planes in three-dimensional space and how they are represented using ordered triples of real numbers called coordinates. It describes how the three mutually perpendicular coordinate axes divide space into eight octants. It provides formulas for finding distances between points, section formulas, midpoints, and centroids. It also discusses direction cosines and direction ratios as ways to represent the direction of lines in space, and provides formulas for finding angles between lines based on their direction cosines or direction ratios.
1. The document introduces analytic geometry and its use of Cartesian coordinate systems to determine properties of geometric figures algebraically.
2. It defines key concepts like directed lines and rectangular coordinates, and explains how to find the distance between two points and the area of polygons using their coordinates.
3. Formulas are provided to calculate distances between horizontal, vertical and slanted line segments, as well as the area of triangles and general polygons from the coordinates of their vertices. Sample problems demonstrate applying these formulas.
Here are the steps to find the line of intersection of the two planes:
1) Write the equations of the planes in standard form:
Plane 1: x + 2y - z = 4
Plane 2: 2x - y + z = 1
2) Set the equations equal to each other and solve as a system of equations:
x + 2y - z = 4
2x - y + z = 1
3) Eliminate one variable:
Subtract the second equation from the first:
(x + 2y - z) - (2x - y + z) = 4 - 1
-x + y = 3
4) Substitute back into one of the
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
This document contains information about graphing algebraic equations:
1) It introduces Cartesian coordinates and using a graph to represent points that satisfy an equation in two variables.
2) It shows that a linear equation represents a straight line on a graph and that simultaneous linear equations have a single intersection point.
3) Quadratic and higher degree equations represent curves on a graph, with the number of intersections with the x-axis equal to the degree of the equation. Intersections can be real, imaginary, or coincident.
4) The absolute term in an equation affects the position but not the shape of its graph. Shifting terms affects the position of intersections along the x-axis.
* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.
1. The document summarizes key concepts from a chapter on coordinate geometry including calculating the distance and midpoint between two points, finding the slope and equation of a line, calculating the area of a triangle, finding the angle between two lines, and dividing a line segment in a given ratio.
2. Formulas are provided for finding the slope, equation, and angle of inclination of a line, as well as the perpendicular distance from a point to a line and the coordinates of the point where two lines intersect.
3. Worked examples demonstrate using the formulas to calculate values like the slope and equation of a line given two points, the area of a triangle using coordinates, and dividing a line segment based on a given
The document discusses coordinate geometry and defines key terms like axes of reference, coordinates of a point, distance formula, section formula, and area of a triangle. It provides examples of using these concepts and formulas to solve problems like finding the coordinates of a point dividing a line segment in a given ratio, or the area of a triangle given the coordinates of its vertices.
The document discusses key concepts of coordinate geometry including: Rene Descartes introduced coordinate geometry by representing points on a plane using two numbers called coordinates; the plane is divided into four parts by perpendicular x and y axes intersecting at the origin; the coordinates of a point P are written as an ordered pair (x,y) giving the distance from the origin along the x and y axes; and the distance formula can be used to find the distance between two points on the plane.
Lecture 1.6 further graphs and transformations of quadratic equationsnarayana dash
1) The graphs of y2 = x and y2 = -x are parabolas with their axis of symmetry along the x-axis rather than the y-axis.
2) If the coordinate axes are rotated by an angle θ, the equation of a parabola changes but the parabola itself remains the same.
3) Rotating the axes transforms the equation Y = X2 into y cosθ - x sinθ = (x cosθ + y sinθ)2, where θ is the angle of rotation.
This document provides a lesson on linear relations and lines. It includes the following key points:
- It introduces the coordinate plane and how points are represented by ordered pairs.
- It discusses relations and functions, including how to represent them using tables, graphs, and mapping diagrams. It also discusses the vertical line test for identifying functions.
- It covers linear relations and how their graphs form straight lines. It shows how to find the slope and equation of a line.
- It examines parallel and perpendicular lines, including how to find the equation of a line parallel or perpendicular to another given line.
03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pp
This document provides information about coordinate grids, ordered pairs, and formulas in coordinate geometry. It defines key terms like coordinates, quadrants, and distance and section formulas. The distance formula calculates the distance between two points with coordinates (x1, y1) and (x2, y2). The section formula finds the coordinates of a point that divides a line segment between (x1, y1) and (x2, y2) in a given ratio. It also discusses finding the midpoint and calculating the area of a triangle using coordinates.
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
The document provides information about analytical geometry and straight lines. It defines key terms like slope, direction cosines, direction ratios, and equations of straight lines. Specifically:
1) Slope is the tangent of the angle between a line and the x-axis. Slope can be calculated as the rise over the run between two points.
2) There are three common forms for the equation of a straight line: slope-intercept form y=mx+c, slope-point form y-y1=m(x-x1), and intercept form x=k.
3) Direction cosines and ratios describe the orientation of a line in 3D space and are proportional to each other. The direction ratios
Module 5 (Part 3-Revised)-Functions and Relations.pdfGaleJean
The Cavite Mutiny of 1872 was an uprising of military personnel of Fort San Felipe, the Spanish arsenal in Cavite, Philippines on January 20, 1872, about 200 Filipino military personnel of Fort San Felipe Arsenal in Cavite, Philippines, staged a mutiny which in a way led to the Philippine Revolution in 1896. The 1872 Cavite Mutiny was precipitated by the removal of long-standing personal benefits to the workers such as tax (tribute) and forced labor exemptions on order from the Governor General Rafael de Izquierdo. Many scholars believe that the Cavite Mutiny of 1872 was the beginning of Filipino nationalism that would eventually lead to the Philippine Revolution of 1896.
The document discusses lines and planes in 3D space. It defines lines as being determined by a point and direction vector, and gives parametric and symmetric equations to represent lines. Planes are defined by a point and normal vector, with standard and general forms for their equations. Methods are provided for finding the intersection of lines or planes, as well as the distance between a point and plane or line. Examples demonstrate finding equations of lines and planes, sketching planes, and determining relationships between lines or planes.
Straight Lines for JEE REVISION Power point presentationiNLUVWDu
The document provides information about the weightage and number of problems on straight lines that have been asked in JEE Main from 2020-2024. It lists important subtopics of straight lines like locus of a point, reflection, pair of lines, etc. It also provides various important formulas related to straight lines, short notes, and sample problems related to important subtopics. The document is a comprehensive reference material for the straight lines chapter of JEE Main exam preparation.
Assignment of straight lines and conic sectionKarunaGupta1982
1. The document contains 35 problems related to straight lines and conic sections like circles, ellipses, parabolas and hyperbolas. The problems involve finding equations of lines and conic sections given certain properties, finding properties like foci, vertices, axes, etc. from given equations.
2. Specific problems include finding slopes of lines, angle between lines, perpendicular lines, finding equations of lines passing through points or perpendicular/parallel to other lines.
3. For conic sections, problems include finding equations given foci, directrix, passing points, intersecting lines, centers and radii of circles, lengths of axes, foci, latus rectum and eccentricities of ellipses and hyperbol
This document discusses coordinates in space and three-dimensional coordinate geometry. It introduces points, lines, and planes in three-dimensional space and how they are represented using ordered triples of real numbers called coordinates. It describes how the three mutually perpendicular coordinate axes divide space into eight octants. It provides formulas for finding distances between points, section formulas, midpoints, and centroids. It also discusses direction cosines and direction ratios as ways to represent the direction of lines in space, and provides formulas for finding angles between lines based on their direction cosines or direction ratios.
1. The document introduces analytic geometry and its use of Cartesian coordinate systems to determine properties of geometric figures algebraically.
2. It defines key concepts like directed lines and rectangular coordinates, and explains how to find the distance between two points and the area of polygons using their coordinates.
3. Formulas are provided to calculate distances between horizontal, vertical and slanted line segments, as well as the area of triangles and general polygons from the coordinates of their vertices. Sample problems demonstrate applying these formulas.
Here are the steps to find the line of intersection of the two planes:
1) Write the equations of the planes in standard form:
Plane 1: x + 2y - z = 4
Plane 2: 2x - y + z = 1
2) Set the equations equal to each other and solve as a system of equations:
x + 2y - z = 4
2x - y + z = 1
3) Eliminate one variable:
Subtract the second equation from the first:
(x + 2y - z) - (2x - y + z) = 4 - 1
-x + y = 3
4) Substitute back into one of the
The document discusses different types of equations that can represent straight lines in a plane, including point-slope form, two-point form, slope-intercept form, and normal form. It provides examples of writing the equation of a line given characteristics like two points, slope and intercept, or being parallel/perpendicular to another line. The document also covers topics like finding the distance from a point to a line and the equations of angle bisectors.
This document contains information about graphing algebraic equations:
1) It introduces Cartesian coordinates and using a graph to represent points that satisfy an equation in two variables.
2) It shows that a linear equation represents a straight line on a graph and that simultaneous linear equations have a single intersection point.
3) Quadratic and higher degree equations represent curves on a graph, with the number of intersections with the x-axis equal to the degree of the equation. Intersections can be real, imaginary, or coincident.
4) The absolute term in an equation affects the position but not the shape of its graph. Shifting terms affects the position of intersections along the x-axis.
* Plot ordered pairs in a Cartesian coordinate system.
* Graph equations by plotting points.
* Find x-intercepts and y-intercepts.
* Use the distance formula.
* Use the midpoint formula.
1. The document summarizes key concepts from a chapter on coordinate geometry including calculating the distance and midpoint between two points, finding the slope and equation of a line, calculating the area of a triangle, finding the angle between two lines, and dividing a line segment in a given ratio.
2. Formulas are provided for finding the slope, equation, and angle of inclination of a line, as well as the perpendicular distance from a point to a line and the coordinates of the point where two lines intersect.
3. Worked examples demonstrate using the formulas to calculate values like the slope and equation of a line given two points, the area of a triangle using coordinates, and dividing a line segment based on a given
The document discusses coordinate geometry and defines key terms like axes of reference, coordinates of a point, distance formula, section formula, and area of a triangle. It provides examples of using these concepts and formulas to solve problems like finding the coordinates of a point dividing a line segment in a given ratio, or the area of a triangle given the coordinates of its vertices.
The document discusses key concepts of coordinate geometry including: Rene Descartes introduced coordinate geometry by representing points on a plane using two numbers called coordinates; the plane is divided into four parts by perpendicular x and y axes intersecting at the origin; the coordinates of a point P are written as an ordered pair (x,y) giving the distance from the origin along the x and y axes; and the distance formula can be used to find the distance between two points on the plane.
Lecture 1.6 further graphs and transformations of quadratic equationsnarayana dash
1) The graphs of y2 = x and y2 = -x are parabolas with their axis of symmetry along the x-axis rather than the y-axis.
2) If the coordinate axes are rotated by an angle θ, the equation of a parabola changes but the parabola itself remains the same.
3) Rotating the axes transforms the equation Y = X2 into y cosθ - x sinθ = (x cosθ + y sinθ)2, where θ is the angle of rotation.
This document provides a lesson on linear relations and lines. It includes the following key points:
- It introduces the coordinate plane and how points are represented by ordered pairs.
- It discusses relations and functions, including how to represent them using tables, graphs, and mapping diagrams. It also discusses the vertical line test for identifying functions.
- It covers linear relations and how their graphs form straight lines. It shows how to find the slope and equation of a line.
- It examines parallel and perpendicular lines, including how to find the equation of a line parallel or perpendicular to another given line.
03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptxV03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pptx03 Parallel and Perpendicular Lines.pp
This document provides information about coordinate grids, ordered pairs, and formulas in coordinate geometry. It defines key terms like coordinates, quadrants, and distance and section formulas. The distance formula calculates the distance between two points with coordinates (x1, y1) and (x2, y2). The section formula finds the coordinates of a point that divides a line segment between (x1, y1) and (x2, y2) in a given ratio. It also discusses finding the midpoint and calculating the area of a triangle using coordinates.
1) The document provides a refresher on analytic geometry concepts including the Cartesian plane, lines, parabolas, ellipses, and circles. It gives definitions, properties, and equations for these concepts.
2) Examples are worked through, such as finding the coordinates of points, slopes of lines, and equations of lines and circles. Practice problems and their solutions are also provided.
3) Key topics covered include the Cartesian plane, distance between points, slope and equations of lines, parallel and perpendicular lines, conic sections including parabolas, circles, and ellipses, and their defining properties and equations.
The document provides information about analytical geometry and straight lines. It defines key terms like slope, direction cosines, direction ratios, and equations of straight lines. Specifically:
1) Slope is the tangent of the angle between a line and the x-axis. Slope can be calculated as the rise over the run between two points.
2) There are three common forms for the equation of a straight line: slope-intercept form y=mx+c, slope-point form y-y1=m(x-x1), and intercept form x=k.
3) Direction cosines and ratios describe the orientation of a line in 3D space and are proportional to each other. The direction ratios
Module 5 (Part 3-Revised)-Functions and Relations.pdfGaleJean
The Cavite Mutiny of 1872 was an uprising of military personnel of Fort San Felipe, the Spanish arsenal in Cavite, Philippines on January 20, 1872, about 200 Filipino military personnel of Fort San Felipe Arsenal in Cavite, Philippines, staged a mutiny which in a way led to the Philippine Revolution in 1896. The 1872 Cavite Mutiny was precipitated by the removal of long-standing personal benefits to the workers such as tax (tribute) and forced labor exemptions on order from the Governor General Rafael de Izquierdo. Many scholars believe that the Cavite Mutiny of 1872 was the beginning of Filipino nationalism that would eventually lead to the Philippine Revolution of 1896.
4th Modern Marketing Reckoner by MMA Global India & Group M: 60+ experts on W...Social Samosa
The Modern Marketing Reckoner (MMR) is a comprehensive resource packed with POVs from 60+ industry leaders on how AI is transforming the 4 key pillars of marketing – product, place, price and promotions.
Predictably Improve Your B2B Tech Company's Performance by Leveraging DataKiwi Creative
Harness the power of AI-backed reports, benchmarking and data analysis to predict trends and detect anomalies in your marketing efforts.
Peter Caputa, CEO at Databox, reveals how you can discover the strategies and tools to increase your growth rate (and margins!).
From metrics to track to data habits to pick up, enhance your reporting for powerful insights to improve your B2B tech company's marketing.
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This is the webinar recording from the June 2024 HubSpot User Group (HUG) for B2B Technology USA.
Watch the video recording at https://youtu.be/5vjwGfPN9lw
Sign up for future HUG events at https://events.hubspot.com/b2b-technology-usa/
Beyond the Basics of A/B Tests: Highly Innovative Experimentation Tactics You...Aggregage
This webinar will explore cutting-edge, less familiar but powerful experimentation methodologies which address well-known limitations of standard A/B Testing. Designed for data and product leaders, this session aims to inspire the embrace of innovative approaches and provide insights into the frontiers of experimentation!
Open Source Contributions to Postgres: The Basics POSETTE 2024ElizabethGarrettChri
Postgres is the most advanced open-source database in the world and it's supported by a community, not a single company. So how does this work? How does code actually get into Postgres? I recently had a patch submitted and committed and I want to share what I learned in that process. I’ll give you an overview of Postgres versions and how the underlying project codebase functions. I’ll also show you the process for submitting a patch and getting that tested and committed.
The Ipsos - AI - Monitor 2024 Report.pdfSocial Samosa
According to Ipsos AI Monitor's 2024 report, 65% Indians said that products and services using AI have profoundly changed their daily life in the past 3-5 years.
1. Geometry is concerned with properties of space that are related with
distance, shape, size, and relative position of figures.
Algebra is the study of mathematical symbols and the rules for
manipulating these symbols appearing in equations describing relationships
between variables.
Rene Descartes coupled the branch of geometry with that of algebra
and thereby the study of geometry could be simplified to a very large extent. Of
course, this hindered the development of abstract geometry, just like the
availability of digital computers blocked the growth of mathematics.
28. Questions (10)
Find the distance between the following pairs of points.
1. 2, 3 𝑎𝑛𝑑 5, 7
2. 1, −7 𝑎𝑛𝑑 −1, 5
3. −3, −2 and −6, 7
4. 𝑎, 0 𝑎𝑛𝑑 0, 𝑏
5. 𝑏 + 𝑐, 𝑐 + 𝑎 𝑎𝑛𝑑 𝑐 + 𝑎, 𝑎 + 𝑏
6. 𝑎 cos 𝛼 , 𝑎 sin 𝛼 𝑎𝑛𝑑 𝑎 cos 𝛽 , 𝑎 sin 𝛽
7. 𝑎 𝑚1
2
, 2𝑎𝑚1 𝑎𝑛𝑑 𝑎 𝑚2
2
, 2𝑎𝑚2
8. Lay down in a figure the positions of the points 1, −3 and −2, 1 and
prove that the distance between them is 5.
9. Find the value of 𝑥1 if the distance between the points 𝑥1, 2 𝑎𝑛𝑑 3, 4
be 8.
10.A line is of length 10 and one end is at the point 2, −3 ; if the abscissa of
the other end be 10, prove that its ordinate must be 3 or -9.
29. Questions (8)
1. Prove that the points 2𝑎, 4𝑎 , 2𝑎, 6𝑎 , and 2𝑎 + 3𝑎, 5𝑎 are the
vertices of an equilateral triangle whose side is 2a.
2. Prove that the points −2, −1 , 1, 0 , 4, 3 , 𝑎𝑛𝑑 1, 2 are at the
vertices of a parallelogram.
3. Prove that the points 2, −2 , 8, 4 , 5, 7 , 𝑎𝑛𝑑 −1, 1 are at the
angular points of a rectangle.
4. Prove that the points −
1
14
,
39
14
is the centre of the circle
circumscribing the triangle whose angular points are 1, 1 , 2, 3 ,
𝑎𝑛𝑑 −2, 2 .
5. Find the coordinates of the point which divides the line joining the points
1, 3 𝑎𝑛𝑑 2, 7 in the ratio 3: 4.
6. Find the coordinates of the point which divides the same line in the ration
3: − 4.
7. Find the coordinates of the point which divides, internally and externally,
the line joining −1, 2 to 4, 5 in the ratio 2: 3.
8. Find the coordinates of the point which divides, internally and externally,
the line joining −3, −4 𝑡𝑜 −8, 7 in the ratio 7: 5.
36. Locus. Equation to a Locus
• Article 36 When a point moves so as always to
satisfy a given condition or conditions, the path it
traces out is called its Locus under these
conditions.
For example, suppose O to be given point in
the plane of the paper and that a point P is to
move on the paper so that its distance from O
shall be constant and equal to a. It is clear that all
the positions of the moving point must lie on the
circumference of a circle whose centre is O and
radius is a. The circumference of this circle is
therefore the “Locus” of P when it moves subject
to the condition that its distance from O shall be
equal to the constant distance a.
37. Equation to a curve
• Definition
The equation to a curve is the relation
which exists between the coordinates of any
point on the curve, and which holds for no
other points except those lying on the curve.
38. Straight Line
Article 46 To find the equation to a straight line
which is parallel to one of the coordinate axes.
Let CL be any line parallel to the axis of y and
passing through a point C on the axis of x such
that OC=c.
Let P be any point this line whose coordinates are x
and y. Then the abscissa of the point P is always c,
so that
x=c ……..(1)
40. Continued
The expression (1) is true for every point on the
line CL (produced indefinitely both ways), and for
no other point , is, by article 42, the equation to
the line.
It should be noted that the equation does not
contain the coordinate y.
Similarly the equation to straight line parallel to the
axis of x is y=d.
Corollary: The equation to the axis of x is y=0.
The equation to the axis of y is x=0.
41.
42. 47. To find the equation to a straight line which cuts off a
given intercept on the axis y and is inclined at a given angle to
the axis of x.
Y L
P
C N
L’
O M X
43. Let the given intercept be c and let the given angle be 𝛼.
Let C be a point on the axis of y such that OC is c. Through C draw a straight line
LCL’ inclined at an angle 𝛼 = tan−1
𝑚 to the axis of x, so that tan𝛼 = 𝑚.
The straight line LCL’ is therefore the straight line required, and we have to find
the relation between the coordinates of any point P lying on it.
Draw PM perpendicular to OX to meet in N a line through C parallel to OX.
Let the coordinates of P be x and y, so that 𝑂𝑀 = 𝑥 𝑎𝑛𝑑 𝑀𝑃 = 𝑦.
Then 𝑀𝑃 = 𝑁𝑃 + 𝑀𝑁 = 𝐶𝑁 tan𝛼 + 𝑂𝐶 = 𝑚𝑥 + 𝑐,
i.e. 𝑦 = 𝑚𝑥 + 𝑐
This relation being true for any point on the given straight line is, by Art 42, the
equation to the straight line.
Corollary: The equation to nay straight line passing through the origin, i.e. which
cuts off a zero intercept from the axis of y, is found by putting 𝑐 = 0 and hence is
𝑦 = 𝑚𝑥.
46. 50. To find the equation to the straight line which cuts off
given intercepts a and b from the axes.
Y
B
P
O M A X
47. Let A and B be on OX and OY respectively, and be such that 𝑂𝐴 = 𝑎 𝑎𝑛𝑑 𝑂𝐵 =
𝑏. Join AB and produce it indefinitely both ways. Let P be any point 𝑥, 𝑦 on this
straight line, and draw PM perpendicular to OX.
We require the relation that always hold between 𝑥 and 𝑦, so long as P lies on AB.
By geometry, we have 𝑠𝑖𝑛𝑐𝑒 ∆𝐵𝑂𝐴 𝑎𝑛𝑑 ∆𝑃𝑀𝐴 𝑎𝑟𝑒 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑠
𝑂𝑀
𝑂𝐴
=
𝑃𝐵
𝐴𝐵
,𝑎𝑛𝑑
𝑀𝑃
𝑂𝐵
=
𝐴𝑃
𝐴𝐵
𝑂𝑀
𝑂𝐴
+
𝑀𝑃
𝑂𝐵
=
𝑃𝐵 + 𝐴𝑃
𝐴𝐵
= 1
i.e.
𝑥
𝑎
+
𝑦
𝑏
= 1
This is therefore the required equation; for it is the relation that holds
between the coordinates of any point lying on the given straight line.
48. 51. Example: Find the equation to the straight line passing through the point
3, −4 and cutting off intercepts, equal but of opposite signs, from the axes.
The equation to the straight line is then
𝑥
𝑎
+
𝑦
−𝑎
= 1
i.e. 𝑥 − 𝑦 = 𝑎 … (1)
Since, in addition, the straight is to go through the point 3, −4 , these
coordinates must satisfy (1), so that
3 − −4 = 𝑎
and therefore 𝑎 = 7.
The required equation is therefore
𝑥 − 𝑦 = 7.
49. 62. To find the equation to the straight line which
passes through the two given points (x’, y’) and (x’’, y’’).
Y
(x’’, y’’)
(x’, y’)
O X
50. By Art. 47, the equation to any straight line is
𝑦 = 𝑚𝑥 + 𝑐 ….(1)
By properly determining the quantities 𝑚 𝑎𝑛𝑑 𝑐 we can make (1)
represent nay straight line we need.
If (1) pass through the point 𝑥′
, 𝑦′ , we have
𝑦′
= 𝑚𝑥′
+ 𝑐 ……(2)
Substituting for 𝑐, from (2), the equation (1) becomes
𝑦 − 𝑦′
= 𝑚 𝑥 − 𝑥′ ……(3)
51. This is the equation to the line going through 𝑥′
, 𝑦′ making an angle
tan−1
𝑚 with OX. If in addition (3) passes through the point 𝑥′′
, 𝑦′′ ,
then
𝑦′′
− 𝑦′
= 𝑚 𝑥′′
− 𝑥′
This yields 𝑚 =
𝑦′′ −𝑦′
𝑥′′ −𝑥′ .
Substituting this value in (3), we get as the required equation
𝑦 − 𝑦′
=
𝑦′′
− 𝑦′
𝑥′′ − 𝑥′
𝑥 − 𝑥′
52. Questions (6)
Find the equation to the straight line
1. Cutting off an intercept unity from the positive direction of the
axis of 𝑦 and inclined at 45° to the axis of 𝑥.
2. Cutting off an intercept −5 from the axis of 𝑦 and being equally
inclined to the axes.
3. Cutting off an intercept 2 from the negative direction of the axis
of 𝑦 and inclined at 30° to OX.
4. Cutting of an intercept −3 from the axis of 𝑦 and inclined at an
angle tan−1 3
5
to the axis of 𝑥.
5. Cutting of intercepts 3 𝑎𝑛𝑑 2 from the axes.
6. Cutting of intercepts −5 𝑎𝑛𝑑 6 from the axes.
53. Questions (4)
1. Find the equation to the straight line which passes through the
point 5, 6 and intercepts on the axes
[1]. Equal in magnitude and both positive,
[2]. Equal in magnitude but opposite in sign.
2. Find the equation to the straight lines which pass through the
point 1, −2 and cut off equal distances from the two axes.
3. Find the equation to the straight line which passes through the
given point 𝑥′
, 𝑦′ and is such that the given point bisects the
part intercepted between the axes.
4. Find the equation to the straight line which passes through the
point −4, 3 and is such that the portion of it between the axes
is divided by the point in the ratio 5 ∶ 3.
55. Questions (6)
Find the equations to the straight lines passing through the following
pairs of points.
1. 𝑎 cos 𝜙1, 𝑎 sin 𝜙1 𝑎𝑛𝑑 𝑎 cos 𝜙2, 𝑎 sin 𝜙2
2. 𝑎 cos 𝜙1, 𝑏 sin 𝜙1 𝑎𝑛𝑑 𝑎 cos 𝜙2, 𝑏 sin 𝜙2
3. 𝑎 sec 𝜙1, 𝑏 tan 𝜙1 𝑎𝑛𝑑 𝑎 sec 𝜙2, 𝑏 tan 𝜙2
Find the equations to the sides of the triangles the coordinates of
whose angular points are respectively.
4. 1, 4 , 2, −3 , 𝑎𝑛𝑑 −1, −2
5. 0, 1 , 2, 0 , 𝑎𝑛𝑑 −1, −2
6. Find the equations to the diagonals of the rectangle the equations
of whose sides are 𝑥 = 𝑎, 𝑥 = 𝑎′
, 𝑦 = 𝑏, 𝑎𝑛𝑑 𝑦 = 𝑏′.
56. 66. To find the angle between two given straight lines
Y
A
C2
C1
L2 L1 O X
57. Let the two straight lines be AL1 and AL2, meeting the axes of 𝑥 𝑖𝑛 𝐿1
and L2. Let their equations be
𝑦 = 𝑚1𝑥 + 𝑐1 𝑎𝑛𝑑 𝑦 = 𝑚2𝑥 + 𝑐2 ……….(1)
We know that
tan𝐴𝐿1𝑋 = 𝑚1,𝑎𝑛𝑑 tan𝐴𝐿2𝑋 = 𝑚2
Now ∠𝐿1𝐴𝐿2 = ∠𝐴𝐿1𝑋 − ∠𝐴𝐿2𝑋.
∴ tan∠𝐿1𝐴𝐿2 = tan ∠𝐴𝐿1𝑋 − ∠𝐴𝐿2𝑋
tan 𝐴𝐿1𝑋−tan 𝐴𝐿2𝑋
1+tan 𝐴𝐿1𝑋∙tan 𝐴𝐿2𝑋
=
𝑚1−𝑚2
1+𝑚1𝑚2
Hence the required angle ∠𝐿1𝐴𝐿2 = tan−1 𝑚1−𝑚2
1+𝑚1𝑚2
….(2)
58. 67. To find the condition that two straight lines may be parallel.
Two straight lines are parallel when the angle between them is zero
and therefore the tangent of this angle is zero. This gives
tan 0° = 0 =
𝑚1 − 𝑚2
1 + 𝑚1𝑚2
; 𝑚1 = 𝑚2
Two straight lines having same 𝑚 will be parallel.
69. To find the condition that two straight lines may be perpendicular.
tan 90° = ∞ =
𝑚1 − 𝑚2
1 + 𝑚1𝑚2
; 1 + 𝑚1 ∙ 𝑚2 = 0; 𝑚1 ∙ 𝑚2 = −1
The straight line 𝑦 = 𝑚1𝑥 + 𝑐1 is therefore perpendicular to
𝑦 = 𝑚2𝑥 + 𝑐2, if 𝑚1 = −
1
𝑚2
.
59. Questions (6)
Find the angles between the pairs of straight lines
1. 𝑥 − 𝑦 3 = 5 𝑎𝑛𝑑 3𝑥 + 𝑦 = 7.
2. 𝑥 − 4𝑦 = 3 𝑎𝑛𝑑 6𝑥 − 𝑦 = 11.
3. 𝑦 = 3𝑥 + 7 𝑎𝑛𝑑 3𝑦 − 𝑥 = 8
4. 𝑦 = 2 − 3 𝑥 + 5 𝑎𝑛𝑑 𝑦 = 2 + 3 𝑥 − 7.
5. Find the tangent of the angle between the lines whose intercepts
on the axes are respectively 𝑎, −𝑏 𝑎𝑛𝑑 𝑏,−𝑎.
6. Prove that the points 2, −1 , 0, 2 , 2, 3 ,𝑎𝑛𝑑 4, 0 are
the coordinates of the angular points of a parallelogram and find
the angle between its diagonal.
60. Questions (4)
Find the equation to the straight line
1. passing through the point 2, 3 and perpendicular to the
straight line 4𝑥 − 3𝑦 = 10.
2. passing through the point −6, 10 and perpendicular to the
straight line 7𝑥 + 8𝑦 = 5.
3. passing through the point 2, −3 and perpendicular to the
straight line joining the points 5, 7 𝑎𝑛𝑑 −6, 3 .
4. passing through the point −4, −3 and perpendicular to the
straight line joining the points 1, 3 𝑎𝑛𝑑 2, 7 .
61. POINT OF INTERSECTION
Find the coordinates of the points of intersection of the straight lines
whose equations are
1. 2𝑥 − 3𝑦 + 5 = 0 𝑎𝑛𝑑 7𝑥 + 4𝑦 = 3.
2.
𝑥
𝑎
+
𝑦
𝑏
= 1 𝑎𝑛𝑑
𝑥
𝑏
+
𝑦
𝑎
= 1.
3. 2𝑥 − 3𝑦 = 1 𝑎𝑛𝑑 5𝑦 − 𝑥 = 3, and the angle at which they cut
one another.
4. 3𝑥 + 𝑦 + 12 = 0 𝑎𝑛𝑑 𝑥 + 2𝑦 − 1 = 0 and the angle at which
they cut one another.
5. Prove that the following sets of three lines meet in a point.
[1]. 2𝑥 − 3𝑦 = 7, 3𝑥 − 4𝑦 = 13, 𝑎𝑛𝑑 8𝑥 − 11𝑦 = 33
[2]. 3𝑥 + 4𝑦 + 6 = 0, 6𝑥 + 5𝑦 = −9, 𝑎𝑛𝑑 3𝑥 + 3𝑦 = −5.
62. The circle: Def. A circle is the locus of a point which moves so that its
distance from a fixed point, called the centre, is equal to a given
distance. The given distance is called the radius of the circle.
Y
P
X
M
O
O
M
63. 139. To find the equation to a circle, having its centre at the origin.
Let O be the centre of the circle and let 𝑎 be its radius.
Let OX and OY be the axes of coordinates. Let P be any point on the
circumference of the circle, and its coordinates be 𝑥 𝑎𝑛𝑑 𝑦.
Draw PM perpendicular to OX and join OP. Then
𝑂𝑀2
+ 𝑀𝑃2
= 𝑎2
i.e. 𝑥2
+ 𝑦2
= 𝑎2
This being the relation which hold between the coordinates of any
point on the circumference is the required equation of the circle.
64. 140. To find the equation to a circle
referred to any rectangular axes.
Y
P
O M N X
C L
65. Let OX and OY be the two rectangular axes.
Let C be the centre of the circle and 𝑎 its radius.
Take any point P on the circumference and draw perpendicular CM and
PN upon OX; Let P be the point 𝑥, 𝑦 .
Let the coordinates of C be ℎ 𝑎𝑛𝑑 𝑘; these are supposed to be known.
We have 𝐶𝐿 = 𝑀𝑁 = 𝑂𝑁 − 𝑂𝑀 = 𝑥 − ℎ,
And 𝐿𝑃 = 𝑁𝑃 − 𝑁𝐿 = 𝑁𝑃 − 𝑀𝐶 = 𝑦 − 𝑘.
Hence, since 𝐶𝐿2
+ 𝐿𝑃2
= 𝐶𝑃2
,
We have 𝑥 − ℎ 2
+ 𝑦 − 𝑘 2
= 𝑎2
This is the required equation.
67. Questions (7)
Find the equation to the circle
1. Whose radius is 3 and whose centre is −1, 2 .
2. Whose radius is 10 and whose centre is −5, −6 .
3. Whose radius is 𝑎 + 𝑏 and whose centre is 𝑎, −𝑏 .
Find the coordinates of the centers and the radii of the circles whose
equations are
1. 𝑥2
+ 𝑦2
− 4𝑥 − 8𝑦 = 41
2. 3𝑥2
+ 3𝑦2
− 5𝑥 − 6𝑦 + 4 = 0
Find the equations to the circles which pass through the points
1. 1, 2 , 3, −4 , 𝑎𝑛𝑑 5, −6
2. 1, 1 , 2, −1 , 𝑎𝑛𝑑 3, 2
68. Acknowledgment
The author acknowledges all the websites
which helped a lot in preparing the slides
presented here meant for B.Sc.(Ag) students.
D C Agrawal
dca_bhu@yahoo.com
Note: There are couple of typographical errors;
you are supposed to find it out.