Question

The line $y=3x$ bisects the angle between the lines $a{x}^2+2axy+y^2=0$ if $a=$

• A. $3$
• B. $11$
• C. $\frac{3}{11}$
• D. $\frac{11}{3}$

Answer 1

The correct option is C $\frac{3}{11}$

Pair of angle bisectors represented by
$a{x}^2+2hxy+b{y}^2=0$ is given by
$\frac{x^2-y^2}{a-b}=\frac{xy}{h}$
$\therefore a{x}^2+2axy+y^2=0$
pair of angle bisector is,
$\frac{x^2-y^2}{a-1}=\frac{xy}{a}$
$a{x}^2-a{y}^2=(a-1)xy$
Given $y=3x$ is one of angle bisector of given lines
$m=\frac{y}{x},$ $a{m}^2+(a-1)x-a=0$
$m=3$ is satisfied to this equation
$9a+3a-3-a=0$
$11a=3$
$\therefore a=\frac{3}{11}$