We will learn how to find the equation of a straight line in twopoint form or the equation of the straight line through two given points.
The equation of a line passing through two points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) is y  y\(_{1}\) = \(\frac{y_{2}  y_{1}}{x_{2}  x_{1}}\)(x  x1)
Let the two given points be (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)).
We have to find the equation of the straight line joining the above two points.
Let the given points be A (x\(_{1}\), y\(_{1}\)), B (x\(_{2}\), y\(_{2}\)) and P (x, y) be any point on the straight line joining the points A and B.
Now, the slope of the line AB is \(\frac{y_{1}  y_{2}}{x_{1}  x_{2}}\)
And the slope of the line AP is \(\frac{y  y_{1}}{x  x_{1}}\)
But the three points A, B and P are collinear.
Therefore, slope of the line AP = slope of the line AB
⇒ \(\frac{y  y_{1}}{x  x_{1}}\) = \(\frac{y_{1}  y_{2}}{x_{1}  x_{2}}\)
⇒ y  y\(_{1}\) = \(\frac{y_{2}  y_{1}}{x_{2}  x_{1}}\) (x  x\(_{1}\))
The above equation is satisfied by the coordinates of any point P lying on the line AB and hence, represents the equation of the straight line AB.
Solved examples to find the equation of a straight line in twopoint form:
1. Find the equation of the straight line passing through the points (2, 3) and (6,  5).
Solution:
The equation of the straight line passing through the points (2, 3) and (6,  5) is
\(\frac{ y  3}{ x + 2}\) = \(\frac{3 + 5}{2  6}\),[Using the form, \(\frac{y  y_{1}}{x  x_{1}}\) = \(\frac{y_{1}  y_{2}}{x_{1}  x_{2}}\)]
⇒ \(\frac{ y  3}{ x + 2}\) = \(\frac{ 8}{ 4}\)
⇒ \(\frac{ y  3}{ x + 2}\) = 2
⇒ y  3 = 2x  4
⇒ 2x + y + 1 = 0, which is the required equation
2. Find the equation of the straight line joining the points ( 3, 4) and (5,  2).
Solution:
Here the given two points are (x\(_{1}\), y\(_{1}\)) = ( 3, 4) and (x\(_{2}\), y\(_{2}\)) = (5,  2).
The equation of a line passing through two points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) is y  y\(_{1}\) = [\(\frac{y_{2}  y_{1}}{x_{2}  x_{1}}\)](x  x\(_{1}\)).
So the equation of the straight line in two point form is
y  y\(_{1}\) = \(\frac{y_{2}  y_{1}}{x_{2}  x_{1}}\) (x  x\(_{1}\))
⇒ y  4 = \(\frac{2  4}{5  (3)}\)[x  (3)]
⇒ y  4 = \(\frac{ 6}{ 8}\)(x + 3)
⇒ y  4 = \(\frac{ 3}{ 4}\)(x + 3)
⇒ 4(y  4) = 3(x + 3)
⇒ 4y  16 = 3x  9
⇒ 3x + 4y  7 = 0, which is the required equation.
11 and 12 Grade Math
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