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.


Given: Points $A( a,\ 3),\ B( 2,\ 1)$ and $C( 5,\ a)$ are collinear. 

To do: To find the value of $a$ and to find the equation of the line.

Solution:

As 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 $BC:( y-1)=-\frac{2}{3}( x-2)$

$\Rightarrow 3y-3=4-2x$

$\Rightarrow 2x+3y=7$

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Updated on: 10-Oct-2022

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