## Different forms of the equation of a straight line

The equation of a straight line can be written in different forms depending on the data given. The different forms are as follows:
•    SLOPE INTERCEPT FORM OF A LINE
•    POINT SLOPE FORM OF A LINE
•    TWO POINT FORM OF A LINE
•    THE INTERCEPT FORM OF A LINE
•    NORMAL FORM OR PERPENDICULAR FORM OF A LINE
•    DISTANCE FORM OF A LINE

Let’s discuss a few of these in detail.

# Slope Intercept Form of a Line

The equation of a line with slope m and making an intercept c on y – axis is y = mx + c

Proof: Let the given line intersects y – axis at B and makes an angle θ with x – axis. Then m = tan θ. Let P (x, y) be any point on the line. Draw PM perpendicular to x – axis and BN perpendicular to PM.

Clearly ∠NBP = θ, BN = OM = x and PN = PM – NM = PM – OB = y – c

From ΔPNB, we have
tan θ = PN/BN = (y – c)/x
m = (y – c)/x
y = mx + c, which is the required equation of the line.

## Important Remarks

1. If the line passes through the origin, then 0 = m0 + c; c = 0. Therefore, the equation of a line passing through the origin is y = mx, where m is the slope of the line.

2. If the line is parallel to x – axis, then m = 0, therefore the equation of a line parallel to x – axis is y = c.

## Point – Slope Form of a Line

The equation of a line which passes through the point (x1, y1) and has the slope m is
y – y1 = m(x – x1)

Proof: Let Q (x1, y1) be the point through which the line passes and let
P (x, y) be any point on the line. Then, slope of the line is y – y1
x – x1
But, m is the slope of the line.
So, m = y – y1
x – x1
y – y1 = m(x – x1)
Hence, y – y1 = m(x – x1) is the required equation of the line.

## Two – Point Form of a line

The equation of a line passing through two points (x1, y1) and (x2, y2) is
y – y1 = y2 – y1    (x – x1)
x2 – x1
Proof: Let m be the slope of the line passing through (x1, y1) and (x2, y2). Then,
m = y2 – y1
x2 – x1
So, the equation of the line is
y – y1 = m (x – x1)   [Using point – slope form]
Substituting the value of m, we obtain
y – y1 = y2 – y1   (x – x1)
x2 – x1
This is the required equation of the line in two point form.

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