Question

Find the equation of bisector of the angle between the lines $4x+3y=7$ and $24x+7y-31=0$. Also find which of them is the bisector of the angle containing origin.

• A. $2x-y-3=0$
• B. $2x+y-3=0$
• C. $x-2y-1=0$
• D. $x-2y+4=0$

Answer 1

The correct option is D $x-2y+4=0$

Equation of angle bisector of lines is

$\left|\frac{A_{1}x+B_{1}y+C_1}{\sqrt{A_1^2+B_1^2}}\right|$ $=\pm \left|\frac{A_{2}x+B_{2}y+C_2}{\sqrt{A_2^2+B_2^2}}\right|$

So,$\left|\frac{4x+3y-7}{\sqrt{4^2+3^2}}\right|=\pm \left|\frac{24x+7y-31}{\sqrt{{\left(24\right)}^2+7^2}}\right|\Rightarrow \left|\frac{4x+3y-7}{5}\right|=\pm \left|\frac{24x+7y-31}{25}\right|$

As $-7$ and $-31$ are of same sign then

Equation of angle bisector containing origin is

$\left|\frac{4x+3y-7}{5}\right|=+\left|\frac{24x+7y-31}{25}\right|\Rightarrow 20x+15y-35=24x+7y-31\Rightarrow 4x-8y+4=0\Rightarrow x-2y+4=0$