**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 XX_{1} and YY_{1}. Place XX_{1} as a horizontal line and YY_{1} as a vertical line.

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

Now,

O is origin.

XX_{1} is called as x -axis.

OX is called as positive directions of x- axis and OX_{1} is negative directions of x-axis.

YY_{1} is called as y- axis.

OY is called as positive directions of y-axis and OY_{1} 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-**

**(i) Quadrant I**

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.

**(ii) Quadrant II**

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.

**(iii) Quadrant III**

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.

**(iv) Quadrant IV**

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**