COORDINATE GEOMETRY
[color=#198f88]This book is intended for assisting students of CBSE India who are in class XI or its equivalent. The purpose is to use the amazing power of GeoGebra applets for discovery based learning by interacting with them. This is reinforced by randomly generated simple questions (for formative assessment and quick feedback)[b][/b] in Numbas developed by Newcastle University.[/color]
Table of Contents
- INTRODUCTION
- Shapes
- POINTS ON A PLANE
- STRAIGHT LINES
- Horizontal Lines
- Vertical lines
- SlopeDefn
- Intuitive Idea of slope
- Slope Types
- Rise & Run
- Slope (Quiz) I
- Slope (Quiz) II
- Slope (Quiz) III
- Collinearity Of Three Points
- General Form
- PointOnALine
- Working With General Form
- Slope Intercept Form
- SlopeInterceptFormPractice
- PointSlopeForm
- PointSlopeFormPractice
- Intercept Form
- interceptFormPractice
- Normal Form
- Parallel lines: Quick Investigation
- Parallel Lines Definition
- Perpendicular Lines
- Perpendicular lines -Definition
- FunWithParallelAndPerpendicularLines
- EXERCISE ON STRAIGHT LINES
- CONIC SECTIONS
- Conic Sections
- CIRCLE
- Equation of a circle
- CircleIntro
- tangentToCircle
- PARABOLA
- Parabola Video
- ParabolaDefn
- How to construct a parabola
- standardParabola
- ParabolaPractice
- ELLIPSE
- On geometric definition of ellipse
- Learning Ellipse
- ExploringEllipse
- ellipsePractice
- HYPERBOLA
- HyperbolaDefinition
- Eccentricity
Shapes
Horizontal Lines
Drag the point A to any location and then drag C along the horizontal line joining A and C. Observe what happens.
Q.1
The blue line is parallel to x-axis. A and C are any two points on it. Then
Q.2
Put A at (-4,2) . Bring C on top of A. Then move C to the right by a distance of 6 units. [br]What will be the coordinates of C?
Q.3
Put A at (-3,3). Bring C on top of it. Then move it to the left a distance of 4 units. What will be the coordinates of C?
Q.4
Supposing A has coordinates (a,b) and C has (c,d). Then we must have
Q.5
What happens when you move C along the line?
Q.6
What is the common feature of all the points on the horizontal line?
EQUATION OF A HORIZONTAL LINE
[size=150][color=#0b5394]Since all the points of a horizontal line have the same y-coordinate which is a constant ,we can write it as a set [/color][math]\left\{\left(x,y\right):y=c\right\}[/math][color=#0b5394] where the constant c is the value of all the y-coordinates. Therefore we define the equation of the line as [/color][math]y=c[/math][color=#0b5394].[/color][/size]
Q.7
What is the equation of line parallel to the x-axis and passing through the point (1,2)?
Q.7
If the point (1,4) lies on the line whose equation is y=c , then
Q.8
The line passing through the points (1,2) and (2,1) is parallel to x-axis . True or False?
Q.9
For what value of [math]\theta[/math] where [math]0<\theta<\pi[/math]the line [math]y=\tan\theta[/math]is not parallel to x-axis ?
Q.10
The point (-1,-3) lies in the region bounded by the lines y=-1 and y=-4. True/False?
Conic Sections
A conic section is the intersection of a plane and a cone. The conic sections circle, ellipse, parabola ,hyperbola and pair of lines can be produced by changing the slope of the plane (that is, the angle between the axes of the cone and the intersecting plane).[br]In the applet below we consider an infinite cone with an angle [math]\alpha[/math].[br][*]Play the animation to see a demonstration of the conics, hyperbola, parabola, ellipse, circle and pair of lines.[br][/*][*]Stop the animation and explore additional special cases by changing the angle of the plane (drag the orange point) and the location of the plane (drag the brown point).[/*]
Equation of a circle
Feel free to move the red and green points and/or the circle anywhere you'd like. Notice something?[br]See a right angled triangle? Does it ring a bell? Assume the distance between the red point and [math]\left(h,k\right)[/math] as [math]r[/math].Can you think of a relation between [math]\left(x,y\right)[/math] and [math]\left(h,k\right)[/math]? [br]
Parabola Video
On geometric definition of ellipse
The applet demonstrates the following:[br][i]An ellipse is the set of all points in the plane, the sum of whose distances to two fixed points (foci) remains constant.[/i][br][list][br][*]Select the length of a piece of string by dragging the endpoints of the blue segment.[br][*]Drag the [b]orange[/b] point to select the position of the focus F1 along [b]the x-Axis or the y-Axis[/b]. The other focus F2 is symmetrical to F1 with respect to the origin.[br][/list][br]A string with the selected length is attached to both foci and is kept tight by the tip of the pencil.[br][list][br][*]Drag the tip of the pencil or press the “Draw” button to trace all points on the plane that satisfy the above definition.[br][*]Hide the pencil by pressing the "Pencil ON/OFF" button; show the ellipse by pressing “Show Ellipse” button, and explore the curve by changing the positions of the foci and the length of the constructing string. [br][*]Bring the two foci to the origin to see the circle as a special case of the ellipse.[br][*]Click on “Labels” to see some terminology.[br][/list]