# If three points (x1, y1) (x2, y2), (x3, y3) lie on the same line, prove that y 2 − y 3 x 2 x 3 + y 3 − y 1 x 3 x 1 + y 1 − y 2 x 1 x 2 = 0 - Mathematics

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

#### Solution

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.

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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 6 Co-Ordinate Geometry
Exercise 6.5 | Q 28 | Page 55