Question

Find the value of $a$ for which the following points $A(a,3),B(2,1)$ and $C(5,a)$ are collinear. Hence find the equation of the line.

Answer 1

Given: $A(a,3),B(2,1)$ and $C(5,a)$ are collinear.
$\therefore$ Slope of $AB=$ Slope of $BC$
$\Rightarrow \frac{1-3}{2-a}=\frac{a-1}{5-2}$
$\Rightarrow \frac{-2}{2-a}=\frac{a-1}{3}$
$\Rightarrow -6=(2-a)(a-1)$
$\Rightarrow -6=2a-2-a^2+a$
$\Rightarrow a^2-3a-4=0$
$\Rightarrow a^2-4a+a-4=0$
$\Rightarrow (a-4)(a+1)=0$
$a=4,-1$
For $a=4$
Slope of $BC=\frac{a-1}{5-2}=\frac{4-1}{3}=\frac{3}{3}=1$
Equation of $BC;(y-1)=1(x-2)$
$\Rightarrow y-1=x-2$
$\Rightarrow x-y=1$

For $a=-1$

Slope of $BC=\frac{a-1}{5-2}=\frac{-1-1}{3}=-\frac{2}{3}$

Equation of $BC:(y-1)=-\frac{2}{3}(x-2)\Rightarrow 3y-3=4-2x$

$\Rightarrow 2x+3y=7$