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

The equations of $PQ$ and $PR$ are given by
$y-1=\frac{-2\mp \tan {45}^{\circ }}{1\pm (-2)\tan {45}^{\circ }}(x-2)$

$\Rightarrow y-1=\left(\frac{-2\mp 1}{1\pm 2}\right)(x-2)$

$\Rightarrow y-1=-\frac{1}{3}(x-2)$ and $y-1=3(x-2)$
$\Rightarrow x+3y=5$ and $3x-y=5$
The combined equation of these two lines is
$(x+3y-5)(3x-y-5)=0$
$\Rightarrow 3x^2-3y^2+8xy-20x-10y+25=0$.