Question

The equation of the pair of perpendicular lines passing through origin and forming an isosceles triangle with the line $2 x+3 y=6$, is

• A. $5 x^{2}-24 x y-5 y^{2}=0$
• B. $4 x^{2}-12 x y-4 y^{2}=0$
• C. $6 x^{2}-5 x y-6 y^{2}=0$
• D. $9 x^{2}+5 x y-9 y^{2}=0$

Answer 1

Answer: (A)

Equation of line $B C$ is $2 x+3 y=6$

Slope of $B C=\frac{-2}{3}$
Slope of $A B=m_{1} \tan 45^{\circ}=\left|\frac{m_{1}+\frac{2}{3}}{1-\frac{2}{3} m_{1}}\right|$
$1-\frac{2}{3} m_{1}=m_{1}+\frac{2}{3} $
$\Rightarrow m_{1}\left(1+\frac{2}{3}\right)=1-\frac{2}{3}$
$\Rightarrow m_{1}=\frac{1}{5}$
$\because$ Line $A B$ and $A C$ are perpendicular
$\therefore m_{2}=-5$
Equation of line $A B$ is $y=\frac{1}{5} x $
$\Rightarrow 5 y-x=0$
Equation of line $A C$ is $y=-5 x $
$\Rightarrow y+5 x=0$
Combining equation is $(5 y-x)(y+5 x)=0$
$\Rightarrow 5 y^{2}+24 x y-5 x^{2}=0$
$ \Rightarrow 5 x^{2}-24 x y-5 y^{2}=0$