Let $P Q R$ be a right angled isosceles triangle, right angled at $P(2,1)$. If the equation of $Q R$ is $2 x+y=3$, then the combined equation of the lines $P Q$ and $P R$ is

• A

  $3 x^{2}-3 y^{2}+8 x y+20 x+10 y+25=0$
• Correct answer
  B

  $3 x^{2}-3 y^{2}+8 x y-20 x-10 y+25=0$
• C

  $3 x^{2}-3 y^{2}+8 x y+10 x+15 y+20=0$
• D

  $3 x^{2}-3 y^{2}-8 x y-10 x-15 y-20=0$

Solution

Let $m$ be the slope of $P Q$.

Slope of $Q R$ is $-2$ $\begin{aligned} &\quad\left|\frac{m+2}{1-2 m}\right|=\tan \frac{\pi}{4}=1 \\ &\Rightarrow \quad m+2=\pm(1-2 m) \Rightarrow m=3,-\frac{1}{3} \end{aligned}$ The lines $P Q$ and $P R$ are $\frac{y-1}{x-2}=3,-\frac{1}{3}$ The combined equation is $(3 x-y-5)(x+3 y-5)=0$ or $3 x^{2}-3 y^{2}+8 x y-20 x-10 y+25=0$

Hence, the correct answer is option: 'B'.