Find the value of $'a'$ if the points $(a,\ 3),\ (6,\ -2)$ and $(-3,\ 4)$ are collinear.

Given: Points $(a,\ 3),\ (6,\ -2)$ and $(-3,\ 4)$ are collinear.

To do: To find the value of $a$.

$( a,\ 3),\ ( 6,\ -2),\ ( -3,\ 4)$ are collinear

Here $x_1=a,\ y_1=3,\ x_2=6,\ y_2=-2,\ x_3=-3,\ y_3=4$

As given that the points are collinear, then, Area of the triangle formed by the given points should be zero.

$\Rightarrow \frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_2-y_1)]$ 

$\Rightarrow a(-2-4)+6(4-3)+3(3+2)=0$

$\Rightarrow a(-6)+6(1)+(-3)\times5=0$

$\Rightarrow -6a+6-15=0$

$\Rightarrow -6a-9=0$

$\Rightarrow a=\frac{9}{-6}=-\frac{3}{2}$