Question

Let $L_1:2x+3y=5$ and $L_2$ be equal sides of a triangle. If $L:x+y=2$ is the perpendicular bisector of the third side, then the equation of $L_2$ is

• A. $3x+y=4$
• B. $x+5y=6$
• C. $x=y$
• D. $3x+2y=5$

Answer 1

The correct option is D $3x+2y=5$


Intersection point of $L_1$ and $L$ is $(1,1)$
Angle between $L$ and $L_2=$ Angle between $L$ and $L_1$
$\Rightarrow \left|\frac{m_2-m}{1+m{m}_2}\right|=\left|\frac{m-m_1}{1+m_{1}m}\right|\Rightarrow \left|\frac{m_2+1}{1-m_2}\right|=\left|\frac{-1+\frac{2}{3}}{1+\frac{2}{3}}\right|\Rightarrow \frac{m_2+1}{1-m_2}=\pm \frac{1}{5}$
$\Rightarrow m_2=\frac{-3}{2},-\frac{2}{3}($same as $L_1)$
So, $m_2=-\frac{3}{2}$

Equation of $L_2$ is
$(y-1)=\frac{-3}{2}(x-1)\Rightarrow L_2:3x+2y=5$