Question

If $m$ is the slope of obtuse angle bisector between the lines $3x-4y+7=0$ and $12x+5y-2=0,$ then the value of $|55m|$ is equal to

Answer 1

Given equations of lines,
$L_1:3x-4y+7=0$
and $L_2:12x+5y-2=0$
$\Rightarrow -12x-5y+2=0$
(making constant term positive)

Now, $a_1{a}_2+b_1{b}_2=3(-12)+(-4)(-5)=-16<0$

Equation of obtuse angle bisector is
$\frac{3x-4y+7}{\sqrt{3^2+(-4{)}^2}}=-\frac{-12x-5y+2}{\sqrt{(-12{)}^2+(-5{)}^2}}$
$\Rightarrow 13(3x-4y+7)=5(12x+5y-2)$
$\Rightarrow 21x+77y-101=0$

Slope, $m=-\frac{21}{77}=-\frac{3}{11}$
$\therefore |55m|=|55\times -\frac{3}{11}|=15$