Class 9 Maths Chapter 3: Coordinate Geometry

Coordinate Geometry is Chapter 3 of NCERT Class 9 Maths. It shows how the position of any point on a flat surface can be described exactly with a pair of numbers, using two perpendicular number lines called axes. These notes cover the Cartesian system, quadrants, sign conventions and plenty of solved examples for quick CBSE revision.

What you will learn

• Why we need two reference lines to locate a point in a plane
• The Cartesian system: x-axis, y-axis, origin and the four quadrants
• The meaning of abscissa and ordinate and how to write coordinates (x, y)
• The sign of coordinates in each quadrant and on the axes
• How to read off the position of a point and find its distance from each axis

Key Concepts and Formulas

Why We Need Coordinates

Saying "the shop is 3 km away" is not enough; you also need a direction. In the same way, a point on a plane can only be fixed when we know its distance from two fixed lines. This idea was developed by the French mathematician and philosopher René Descartes, which is why the system is called the Cartesian system.

• A single number line can fix a point on a line, but a plane needs two numbers.
• The two numbers are always measured from two perpendicular lines.

The Cartesian Plane

Draw a horizontal number line and a vertical number line that cross each other at their zero points. Together they divide the plane into four parts.

x-axis: horizontal line X′OX
y-axis: vertical line Y′OY
Origin O = (0, 0)

• On the x-axis, numbers to the right of O are positive and to the left are negative.
• On the y-axis, numbers above O are positive and below O are negative.
• The plane containing the axes is called the Cartesian plane or coordinate plane.

Abscissa and Ordinate

Every point P in the plane is written as an ordered pair (x, y). The order matters, so we always write the x-value first.

Point P = (x, y)
x = abscissa = distance from the y-axis
y = ordinate = distance from the x-axis
Distance of P from x-axis = |y|
Distance of P from y-axis = |x|
(x, y) ≠ (y, x) when x ≠ y

• The abscissa is measured along the x-axis and the ordinate along the y-axis.
• For example, (2, 7) and (7, 2) are different points.

Quadrants and Sign Conventions

The axes split the plane into four quadrants, numbered anticlockwise starting from the top right.

Quadrant I:   (+, +)  e.g. (3, 5)
Quadrant II:  (−, +)  e.g. (−3, 5)
Quadrant III: (−, −)  e.g. (−3, −5)
Quadrant IV:  (+, −)  e.g. (3, −5)

Points on the Axes

A point lying on an axis does not belong to any quadrant.

Point on x-axis: (x, 0)
Point on y-axis: (0, y)
Origin: (0, 0), on both axes

• A point on the x-axis has ordinate zero; a point on the y-axis has abscissa zero.

Solved Examples

Example 1: In which quadrant does the point (−4, 7) lie?

Abscissa = −4 (negative)
Ordinate = 7 (positive)
Signs (−, +)
∴ (−4, 7) lies in Quadrant II

Example 2: Name the quadrants of (5, −2) and (−3, −6).

(5, −2): signs (+, −) → Quadrant IV
(−3, −6): signs (−, −) → Quadrant III
∴ Quadrant IV and Quadrant III respectively

Example 3: Where does the point (0, −9) lie?

Abscissa = 0, so the point is on the y-axis
Ordinate = −9, so it is below the origin
∴ On the y-axis, 9 units below the origin

Example 4: Write the abscissa and ordinate of P(−6, 2) and its distances from the axes.

Abscissa = −6, ordinate = 2
Distance from x-axis = |2| = 2 units
Distance from y-axis = |−6| = 6 units
∴ 2 units from x-axis, 6 units from y-axis

Example 5: A point lies on the x-axis, 4 units to the left of the origin. Find its coordinates.

On x-axis, so ordinate = 0
Left of origin, so abscissa = −4
∴ Point = (−4, 0)

Example 6: A point has ordinate 3 and abscissa −5. Write its coordinates and quadrant.

(x, y) = (−5, 3)
Signs (−, +)
∴ (−5, 3) lies in Quadrant II

Example 7: OABC is a rectangle in the first quadrant with O at the origin, A on the x-axis with OA = 6 units and C on the y-axis with OC = 4 units. Find B.

A = (6, 0), C = (0, 4)
B is 6 units from the y-axis, so x = 6
B is 4 units from the x-axis, so y = 4
∴ B = (6, 4)

Common Mistakes to Avoid

• Never swap the order: the x-coordinate comes first. Writing (y, x) gives a completely different point.
• Do not assign a quadrant to points like (0, 5) or (−2, 0); they lie on an axis, not in a quadrant.
• The abscissa is the distance from the y-axis, not from the x-axis. Many students mix these up.
• Remember quadrants are numbered anticlockwise, so Quadrant II is top left, not bottom right.

Quick Revision

Origin = (0, 0)
Signs: I (+, +), II (−, +), III (−, −), IV (+, −)
Point on x-axis: (x, 0)
Point on y-axis: (0, y)
(x, y) = (y, x) only when x = y

• Abscissa is the x-coordinate; ordinate is the y-coordinate.
• Points on the axes do not lie in any quadrant.

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