Question

If three points (x₁, y₁) (x₂, y₂), (x₃, y₃) lie on the same line, prove that  \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\]

 

Answer 1

GIVEN: If three points (x₁, y₁) (x₂, y₂) and (x₃, y₃)  lie on the same line

TO PROVE:  \[\frac{y_2 - y_3}{x_2 x_3} + \frac{y_3 - y_1}{x_3 x_1} + \frac{y_1 - y_2}{x_1 x_2} = 0\]

PROOF:

We know that three points (x₁, y₁) (x₂, y₂) and (x₃, y₃)   are collinear if

`x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 ) = 0`

⇒ `x_1 (y_2 - y_3) + x_2 (y_3 - y_1) + x_3 (y_1 - y_2 ) = 0`

Dividing by `x_1 x_2 x_3`

⇒   \[\frac{x_1 (y_2 - y_3 ) }{x_1 x_2 x_3} + \frac{x_2 (y_3 - y_1 ) }{x_1x_2 x_3} + \frac{x_3 ( y_1 - y_2 ) }{x_1 x_2 x_3} = 0\]

⇒ \[\frac{(y_2 - y_3)}{x_2 x_3} + \frac{(y_3 - y_1)}{x_3 x_1} + \frac{(y_1 - y_2)}{x_1 x_2} = 0\]

Hence proved.