Coordinate Geometry for Class 9

The French mathematician René Descartes in 1637 introduced the cartesian system of coordinates for describing the position of a point in a plane. This idea has given rise to an important branch of mathematics known as Coordinate Geometry. Coordinate geometry is also known as cartesian geometry. It is a branch of mathematics that merges algebraic techniques with geometric insights. In this chapter, we will explore the fundamentals of coordinate geometry, its key concepts, and practical applications.

Coordinate Geometry

The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry.

Coordinate System

A coordinate system is a way of describing the position of points in space using numbers. It typically involves two axes that intersect at a point called the origin. In a two-dimensional coordinate system, you have two axes (usually labelled x and y).

Points on the plane or in space are located using numerical coordinates which represent distances along each axis from the origin. The coordinates are often written as ordered pairs in two-dimension.

Cartesian Coordinates

Numbers that indicate the location of a point relative to an origin are its shortest distances from two fixed axes that intersect at right angles (perpendicular) at the origin.

Coordinate Axes

The position of a point in a plane is determined by two fixed perpendicular lines called the coordinate axes.
Draw two lines X′OX and YOY′ which are perpendicular to each other and intersect at point O. These lines are called the coordinate axes.
The horizontal line X′OX is called the x-axis.
The vertical line YOY′ is called the y-axis.
The point O is called the origin.
The axes and origin are shown in the graph.

Ordered Pair

A pair of numbers a and b listed in a specific order with ‘a’ in the first place and ‘b’ in the second place is called an ordered pair (a, b). 

Note: (a, b) ≠ (b, a)

Thus, (2, 3) is one ordered pair, and (3, 2) is another ordered pair.

Convention of Signs

The distances measured along OX and OY are taken as positive and those along OX′ and OY′ are taken as negative, as shown in the figure given below.

Coordinates of a point in a Plane

P is a point in a plane and the distance of P from the y-axis is a. And, the distance of P from the x-axis is b. Hence, the coordinates of P are (a, b).

a is called the x-coordinate or abscissa of P.
b is called the y-coordinate or ordinate of P.
The coordinates of the origin are (0, 0).

Quadrants

In a coordinate system, the coordinate plane is divided into four regions known as quadrants. X′OX and YOY′ axes divide the plane of the paper into four regions called quadrants. The regions XOY, YOX′, X′OY′ and Y′OX are known as the first, second, third and fourth quadrants.
Note: Any point on the x-axis or y-axis is not in any quadrant.

Using the convention of signs, we have the signs of the coordinates in various quadrants as given below.

Example:

In which quadrants do the given points lie?

(i) (3, 5)
(ii)(−3, 7)
(iii) (−7, −9)
(iv) (5, −7)

Explantion:

(i) Point (3, 5) of the type (+, +) lies in the 1st quadrant. Hence, point (3, 5) lies in quadrant I.
(ii) Point (−3, 7) of the type (−, +) lies in the 2nd quadrant. Hence, point (−3, 7) lies in quadrant II.
(iii) Point (−7, −9) of the type (−, −) lies in the 3rd quadrant. Hence, point (−7, −9) lies in quadrant III.
(iv) Point (5, −7) of the type (+, −) lies in the 4th quadrant. Hence, point (5, −7) lies in quadrant IV.

Points on Coordinate Axes

(i) Coordinates of a point on the X-axis

Every point on the x-axis is at a distance of 0 units from the x-axis. So, its ordinate is 0. Thus, the coordinates of every point on the x-axis are of the form (x, 0).

(ii) Coordinates of a point on the Y-Axis 

Every point on the y-axis is at a distance of 0 units from the y-axis. So, its abscissa is 0. Thus, the coordinates of every point on the y-axis are of the form (0, y).

Example:

On which axes do the given points lie?

(i) (6, 0)
(ii) (0, 3)

Explantion:

(i) In point (6, 0), the ordinate is 0. Thus, point (6, 0) lies on the x-axis.

(ii) In point (0, 3), the abscissa is 0. Thus, point (0, 3) lies on the y-axis.

Points on Coordinate axes and quadrants are shown as: