Coordinate Geometry

Notes of chapter: Coordinate Geometry are presented below. Indepth notes along with worksheets and NCERT Solutions for Class 9.

(1)Coordinate geometry

The branch of Mathematics in which we study to find out location of a point on a plane is known as coordinate geometry. This branch is developed by French philosopher and mathematician Rene Descartes.

Examples of real life

(i) Finding a house according to street plane.

(ii) Seating plan in cinema halls, class rooms etc.

(iii) Finding shops in mall and market.

(2) Cartesian System

(i) The point from which the distances are marked on number line is called the origin.

O is origin.

(ii) Take two number lines XX1 and YY1. Place XX1 as a horizontal line and YY1 as a vertical line.

(iii) Place these two lines in such a way that their zeroes or origin will meet.

Now,

O is origin.

XX1 is called as x -axis.

OX is called as positive directions of x- axis and OX1 is negative directions of x-axis.

YY1 is called as y- axis.

OY is called as positive directions of y-axis and OY1 is called as negative directions of y – axis. These axes are called coordinate axes.

(iv) These horizontal and vertical lines divide the plane into four parts which are called quadrants. These quadrants are represented by numbers as I, II, III and IV. The plane is called as the Cartesian plane or the coordinate plane or the xy plane.

Signs of the coordinates of a point in different quadrants-

A point lies under positive x-axis and y-axis.

Therefore, point has both positive signs which can represent as (+, +).

Eg:-  Show point A(2,3) on the coordinate plane.

A point lies under negative x-axis and positive y-axis.

Therefore, point has negative sign for x coordinates and positive sign for y coordinates which can represent as (-, +).

Eg:-  Show point A(-2,3) on the coordinate plane.

A point lies under negative x-axis and negative y-axis.

Therefore, point has both negative signs for x coordinates and y coordinates which can represent as (-, -).

Eg:-  Show point A(-2,-3) on the coordinate plane.

A point lies under positive x-axis and negative y-axis.

Therefore, point has positive sign for x coordinates and negative sign for y coordinates which can be represent as (+, -).

Eg:-  Show point A(2,-3) on the coordinate plane.

(3) Steps to write coordinates of a point –

Every point has two coordinates. One represents x – axis coordinate and other represents y- axis coordinate. Both coordinates placed in brackets. x-coordinate comes first and then y- coordinate. The x-coordinate is also called abscissa and y- coordinate is called as ordinate.

Eg:- Write x and y coordinates of the point P(4,3) and point Q(-6, -2).

P(4,3)

x -coordinate is 4

y – coordinate is 3.

Q(-6,-2)

x-coordinate is -6

y -coordinate is -2.

Steps to write x and y coordinates –

The x – coordinate of a point P is its perpendicular distance from the y -axis measured along the x-axis according to positive or negative direction of the x -axis.

The y -coordinate of a point is its perpendicular distance from the x – axis measured along the y-axis according to positive or negative direction of the y -axis.

Eg 1:- Write the coordinates of point P in diagram below-

Ans-

The x – coordinate of a point P is its perpendicular distance from the y -axis measured along the x-axis is PN = OM = 4 units.

The y -coordinate of a point is its perpendicular distance from the x – axis measured along the y-axis is PM = ON = 3 units.

Hence, coordinates of P are (4, 3).

Eg 2:- Write the coordinates of point Q in diagram below-

Ans-

The x – coordinate of a point Q is its perpendicular distance from the y -axis measured along the x-axis is QS = OR = -6 units.

The y -coordinate of a point is its perpendicular distance from the x – axis measured along the y-axis is QR = OS = -2 units.

Hence, coordinates of Q are (-6, -2).

(4) Coordinates of origin O is always zero for both axis because its coordinate for x axis (abscissa) and coordinates for y axis (ordinate) both are zero. Hence coordinates of the origin is (0, 0).

(5) Steps to plot a point in the plane if its coordinates are given-

Eg :- Plot a point A (3, 2) on the plane.

Ans-

Step 1

Draw x-axis and y-axis which cross at zero.

Step 2

To locate point A, move according to sign of the coordinates. In given case, both the signs are positive, so point will be in quadrant I.

Now, move 3 units on x -axis from origin in quadrant I and mark it as M.

Step 3

Move 2 units on y axis from point M in quadrant I (as our point A lays in quadrant I). Mark, point as A. it is our required point.

Eg:- Show points A(4,5), B(-4, 5), C(-4, -5), D(4, -5)

Ans-

(These points will lie in different quadrants. It can be find easily by their sign.)

Point A will be plotted in quadrant I because x and y coordinates are positive.

Point B will be plotted in quadrant II because x coordinate is negative and y coordinate is positive.

Point C will be plotted in quadrant III because x and y coordinates are negative.

Point D will be plotted in quadrant IV because x coordinate is negative and y coordinate is positive.

Taking 1 cm = 1 unit, we draw the x- axis and the y- axis. The positions of the points are shown in diagram below-

(6) If x  y, then the position of (x, y) in the Cartesian plane is different from the position of (y, x).

Eg:- Show coordinate (4,3) and (3,4)have different positions.

Ans-

Let point A has coordinate (4, 3) and point B has coordinate (3, 4).

Both points have positive coordinates. Therefore they will be in quadrant I.

Taking 1 cm = 1 unit, we draw the x- axis and the y- axis. The positions of the points are shown in diagram below-

Therefore, point A and point B have different positions.

(7) If x = y, then the position of (x, y) in the Cartesian plane is same as the position of (y, x).

Eg:- Show coordinate point A(3,3) and Point B(3,3)have same positions.

Both points have positive coordinates. Therefore they will lie in quadrant I.

Taking 1 cm = 1 unit, we draw the x- axis and the y- axis. The positions of the points are shown in diagram below-

Therefore, point A and point B have same positions.

Helping Points

NCERT Solutions Class 9

Worksheet Class 9