Question

If the points $A(-1,-4),B(b,c)$ and $C(5,-1)$ are collinear and $2b+c=4$, find the values of $b$ and $c$.

Answer 1

Given, $A(-1,-4),B(b,c)andC(5,-1)$

Two points are said to be collinear if their slopes are equal

$\Rightarrow$ Slope of $AB$ $=$Slope of $BC$

We have,

Slope between two points$=\left(\frac{y_2-y_1}{x_2-x_1}\right)$

$\Rightarrow$ $\frac{c+4}{b+1}=\frac{-1-c}{5-b}$

$\Rightarrow$ $(c+4)(5-b)=(-1-c)(b+1)$

As we know, $2b+c=4$ $\Rightarrow$ $c=4-2b$

Substituting this value, we get
$\Rightarrow$ $(4-2b+4)(5-b)=(-1-4+2b)(b+1)$

$\Rightarrow$ $(8-2b)(5-b)=(2b-5)(b+1)$

$\Rightarrow$ $40-8b-10b+2b^2=2b^2+2b-5b-5$

$\Rightarrow$ $10-18b=-3b-5$

$\Rightarrow$ $-18b+3b=-5-40$

$\Rightarrow$ $-15b=-45$

$\Rightarrow$ $b=3$ and $c=-2$