Question

If the points $A(-2,1),B(a,b)$ and $C(4,-1)$ are collinear and $a-b=1$, find the values of $a$ and $b$.

Answer 1

If three points $A,B,C$ are collinear

Slope of AB $=$ Slope of BC

Slope of AB $=\frac{b-1}{a+2}$

Slope of AC $=\frac{-1-1}{4+2}=-\frac{2}{3}$

Since $A,B,C$ are collinear

$\frac{b-1}{a+2}=-\frac{2}{3}\Rightarrow 3b-3+2a-4=0$

$3b+2a=7$ ............. $\left(1\right)$

Given $a-b=1\Rightarrow 3a-3b=3$ ........... $\left(2\right)$

Adding both $\left(1\right)$ and $\left(2\right)$, we get

$3b-3b+3a+2a=7+3$

$5a=10\Rightarrow a=2$

Substituting it in equation $\left(2\right)$, we get

$2-b=1\Rightarrow b=1$

Hence $a=2,b=1$.