Straight Lines, Two Point Form, Intercept Form, Perpendicular Form (Normal Form)

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(C) Two Point Form

Let and be two given distinct points.

Slope of the line passing through these points is given by

From the equation of line in point slope form, we get

Which is the required equation of the line in two-point form.

Example:

Find the equation of the line passing through and .

Solution:

The equation of a line passing through two points and is given by

Since and , and , equation (1) becomes,

Or

Or

(D) Intercept Form

We want to find the equation of a line which cuts off given intercepts on both the co-ordinate axes.

Intercept Form

Intercept Form

Intercept Form

Let be a line meeting x-axis in and y-axis in . Let .

Then the co-ordinates of and are and respectively.

The equation of the line joining and is

or

or

or

This is the required equation of the line having intercepts and on the axes.

Example:

Find the equation of a line which cuts off intercepts and on x and y axes respectively.

Solution:

The intercepts are and on x and y axes respectively. i.e.,

The required equation of the line is

(E) Perpendicular Form (Normal Form)

We now derive the equation of a line when p be the length of perpendicular from the origin on the line and , the angle which this perpendicular makes with the positive direction of x-axis is are given.

Perpendicular Form

Perpendicular Form

Perpendicular Form

(i) Let be the given line cutting off intercepts and on x–axis and y –axis respectively.

Let be perpendicular from origin on and (See Fig.(i))

And,

The equation of line is

or

(ii) (From Fig. (ii))

Similarly,

The equation of the line is or

Note:

1. is the length of perpendicular from the origin on the line and is always taken to be positive.

2. is the angle between positive direction of x-axis and the line perpendicular from the origin to the given line.

Example:

Find the equation of the line whose perpendicular distance from the origin is units and the perpendicular from the origin to line makes an angle of with the positive direction of x-axis.

Solution:

Here

The equation of the line is,

or

or

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