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
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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Reference Links :
- http://en.wikipedia.org/wiki/Linear_equation#General_form
- http://en.wikipedia.org/wiki/Slope
- http://en.wikipedia.org/wiki/Intercept
- http://en.wikipedia.org/wiki/Perpendicular