Question

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

• A. $3x^2-3y^2+8xy+20x+10y+25=0$
• 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 the slopes of $PQ$ and $PR$ be $m$ and $\frac{-1}{m}$ respectively. Since $PQR$ is an isosceles triangle $\angle PQR=\angle PRQ$

$\Rightarrow \left|\frac{m+2}{1-2m}\right|=\left|\frac{-\frac{1}{m}+2}{1+\frac{2}{m}}\right|$
[$\because$ slope of QR = -2]
$\Rightarrow m+2=\pm (1-2m)\Rightarrow m=3or-\frac{1}{3}$
So the equations of PQ and PR are
$(y-1)$ = $3(x-2)$ and $y-1=(-\frac{1}{3})(x-2)$