Question

For what value of $x$ are the points $(1,5),(2,3)$ and $(x,x)$ collinear?

Answer 1

Area of the triangle formed by given points

$=\frac{1}{2}|1(3-x)+2(x-5)+x(5-3)|$

$=\frac{1}{2}|3-x+2x-10+2x|$

$=\frac{1}{2}|3x-7|$

The given points are collinear if and only if area of triangle$=0$

$\Rightarrow \frac{1}{2}|3x-7|=0$

$\Rightarrow (3x-7)=0$

$\Rightarrow x=\frac{7}{3}$

$\therefore$ the required value of $x=\frac{7}{3}$