Question

Given vertices $A(1,1),B(4,-2)$ and $C(5,5)$ of a triangle, If the equation of the perpendicular dropped from $C$ to the interior bisector of the angle $A$ is $y=mx-30$.Find the value of $m$.

Answer 1

Equation of AC-

$(y-1)=\frac{5-1}{5-1}(x-1)$

$\Rightarrow y=x$

Equation of AB-

$(y-1)=\frac{-2-1}{4-1}(x-1)$

$\Rightarrow y=-x+2$

$\because m_1.m_2=-1$

$\therefore$ AB and AC are perpendicular to each other.

Hence equation of perpendicular dropped from C to to the angle bisector will be BC itself.

$\therefore$ Equation of BC-

$(y-(-2))=\frac{5-(-2)}{5-4}(x-4)$

$\Rightarrow y=7x-30$

Comparing with the given equation, i.e., $y=mx-30$, we get

$m=7$

Hence the correct answer is $7$.