Question

Let $PQR$ be a right angled isosceles traingle, right angled at $P(2,1).$ If the equation of the line $QR$ is $2x+y=3$, then the equation represnting the pair of lines $PQ$ and $PR$ is

• A. $3x^2-3y^2+8xy+2x+10y+25=0$
• B. $3x^2-3y^2+8xy-20x-10y+25=0$
• C. $3x^2-3y^2+8xy+10x+15y+20=0$
• D. $3x^2-3y^2-8xy-10x-15y-20=0$

Answer 1

The correct option is B $3x^2-3y^2+8xy-20x-10y+25=0$

Let $m$ be the slope of $PQ$ them

$\tan {45}^o=\left|\frac{m-(-2)}{1+m(-2)}\right|$

$\Rightarrow 1=\left|\frac{m+2}{1-2m}\right|\Rightarrow \pm 1=\frac{m+2}{1-2m}$

$\Rightarrow m+2=1-2m$ or $-1+2m=m+2$

$m=-\frac{1}{3}$ or $m=3$

$PQ$ markes ${45}^o$ with $QR$

Equation of $PQ$ is

$y-1=-\frac{1}{3}(x-2)$ $\Rightarrow x+3y-5=0$

Equation of $PR$ is

$y-1=3(x-2)$ $\Rightarrow 3x-y-5=0$

Combined equation of $PQ$ and $PR$ is

$(x+3y-5)(3x-y-5)=0$

$\Rightarrow 3x^2-3y^2+8xy-20x-10y+25=0$