Chapter Notes: Co-ordinate Geometry

What is a Coordinate System?

A Coordinate System is a mathematical framework used to determine the position or location of points in space. It provides a way to describe the position of objects or points using numerical values called coordinates.

• As shown in the figure, the line XOX′ is known as the X-axis, and YOY′is known as the Y-axis.
• The point O is called the origin. For any point P (x y), the ordered pair(x,y) is called the coordinate of point P.
• The distance of a point from the Y-axis is called its abscissa and the distance of a point from the X-axis is called its ordinate.

Distance between Two Points Using Pythagoras' Theorem

Let P(x₁, y₁) and Q(x₂, y₂) be any two points on the cartesian plane.

Draw lines parallel to the axes through P and Q to meet at T.

ΔPTQ is right-angled at T.

By Pythagoras Theorem,

PQ² = PT² + QT²

= (x₂ - x₁)² + (y₂ - y₁)²

PQ = √[x₂ - x₁)² + (y₂ - y₁)²]

Distance Formula

Distance between any two points (x₁, y₁) and (x₂, y₂) is given by

d = √[x₂ - x₁)²+(y₂ - y₁)²]

Where d is the distance between the points (x₁,y₁) and (x₂,y₂).

Note:The distance of any point P(x,y) from the origin O(0,0) is given by:

Example 1: Find the distance between the points D and E, in the given figure.

Solution:

Example 2: What is the distance between two points (2, 3) and (-4, 5) using the distance formula?

Sol: The distance formula is used to calculate the distance between two points in a coordinate plane. It is given as:

d = √[(x2 - x1)² + (y2 - y1)²]

Using this formula, we can find the distance between the points (2, 3) and (-4, 5) as follows:

d = √[(-4 - 2)² + (5 - 3)²]

d = √[(-6)² + (2)²]

d = √[36 + 4]

d = √40

d = 6.32 (approx.)

Therefore, the distance between the points (2, 3) and (-4, 5)is approximately 6.32 units.

Section Formula

Let P (x,y) be a point on the line segment joining A(x₁, y₁) and B(x₂, y₂) such that it divides AB internally in the ratio m:n. The coordinates of the point P are given by
This is known as the Section Formula.

Note:

(i) If the point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio k:1, its coordinates are given by:

Example 2: In what ratio does the point (2,- 5) divide the line segment joining the points A(-3, 5) and B(4, -9)?

Sol: Let the ratio be λ : 1

We have put m = λ and n = 1
or

But, coordinates of point is given as p(2,-5) 

But, coordinates of point is given as p(2,-5) 

4λ - 3 = 2(λ + 1)
⇒ 4λ = 2λ + 2 + 3

⇒ 2λ = 5
⇒ λ = 5/2

The required ratio is 5:2.

Mid -Point Formula

The mid-point of the line joining A(x₁, y₁) and B(x₂, y₂) is given as

Example 3:Suppose we have two points A(2, 4) and B(6, 8). We want to find the midpoint of the line segment AB.

Sol:Using the midpoint formula:

Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

= ((2 + 6) / 2, (4 + 8) / 2)

= (8 / 2, 12 / 2)

= (4, 6)

Therefore, the midpoint of the line segment AB is M(4, 6).

Some Solved Questions

Q1: Find the distance between the points (3, 5) and (-2, -1) using the distance formula.

Sol:

Using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

Substituting the coordinates:

d = √[(-2 - 3)² + (-1 - 5)²]

d = √[(-5)² + (-6)²]

d = √[25 + 36]

d = √61

Therefore, the distance between the points (3, 5) and (-2, -1) is √61 units.

Q2: Find the coordinates of the midpoint of the line segment joining the points (-3, 2) and (5, -4).

Sol:

Using the midpoint formula:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Substituting the coordinates:

Midpoint = ((-3 + 5) / 2, (2 + (-4)) / 2)

Midpoint = (2 / 2, -2 / 2)

Midpoint = (1, -1)
Therefore, the midpoint of the line segment joining (-3, 2) and (5, -4) is (1, -1).