We are already familiar with plotting a point on a plane graph paper. For this we take two perpendicular lines XoX’ and YoY’ intersecting at O. XOX’ is called x-axis or abscissa and YoY’ is called y-axis or ordinate.

Point in a plane

Let us take a point P in a plane. Let XOX’ and YOY’ be pendicualr to each other at O. are drawn. If OM = x and ON = y then x-coordinate of P is x and y-coordinate of P is y. Here we write x-coordinate first. Hence (x, y) and (y, x) are different point whenever .

The two lines XOX’ and YOY’ divides the plane into four parts called quadrants. XOY, YOX’, X’OY’ and Y’OX are respectively the first second, third and and fourth quadrants. We take the direction from O to X and O to Y as positive and the direction from O to X’ and O to Y’ as negative.

Distance between two points

Let P (x₁, y₁) and Q (x₂, y₂) be the two points. We have to find PQ.

OM = x₁, PM = y₁ = RN

ON = x₂, QN = y₂

PR = MN = ON – OM

= x₂ – x₁

QR = QN – RN = y₂ – y₁

By Pythagoras theorem

PQ² = PR² + QR²

= (x₂ - x₁)² + (y₂ – y₁)²

If x₁ = 0, y₁ = 0, x₂ = x and y₂ = y

Then

Section Formula

Let P (x, y) divided a line AB such that AP : PB = m₁ : m₂.

Let coordinates of A are (x₁, y₁) and B are (x₂, y₂).

It is obvious that

Taking,

Similarity

Note

(i) if P is mid point of AB, then AP : PB = 1 : 1

(ii) If m₁ : m₂ = k, then coordinates of P are