# Straight Lines, Conversion into Perpendicular Form, Distance of a Given Point a Given Line

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## Conversion into Perpendicular Form

Suppose the given first degree equation in and is

We will convert this general equation in perpendicular form. For this purpose let us re-write the given equation (1) as

Multiplying both sides of the above equation by , we have

Let us choose such that

Or (Taking positive sign)

Substituting this value of in (2), we have

This is required conversion of (1) in perpendicular form. Two cases arise according as is negative or positive.

(i) If , the equation (2) is the required form.

(ii) If , the R. H. S. of the equation of (3) is negative.

We shall multiply both sides of the equation of (iii) by .

The required form will be

Thus, length of perpendicular from the origin

Inclination of the perpendicular with the positive direction of x-axis is given by

or

Where the upper sign is taken for and the lower sign for 0. If , the line passes through the origin and there is no perpendicular from the origin on the line.

Example:

Reduce the equation into perpendicular form.

Solution:

The equation of given line is

Comparing (1) with general equation of straight line, we have, and

Dividing equation (1) by 2, we have

Or

Or

(and being both negative in the third quadrant, value of θ∴will lie in the third quadrant).

This is the representation of the given line in perpendicular form.

## Distance of a Given Point a Given Line

In this section, we shall discuss the concept of finding the distance of a given point from a given line or lines.

Let be the given point and be the line .

Let the line intersect x axis and y axis and respectively.

Draw and let .

Let the coordinates of be

lies on , or

The coordinates of and are and respectively.

The slope of and the slope of

As or

From

(Using properties of Ratio and Proportion)

Also

From and , we get

Or [Using]

Or [Using]

Since the distance is always positive, we can write

Example:

Find the points on the x-axis whose perpendicular distance from the straight line is .

Solution:

Let be any point on x-axis.

Equation of the given line is . The perpendicular distance of the point from the given line is,

Thus, the point on x-axis is