Question

If the angle between the pair of straight lines formed by joining the points of intersection of $x^2+y^2=4$ and $y=3x+c$ to the origin is right angle, then $c^2=$

Answer 1

$x^2+y^2=4\to \left(1\right)$
$\frac{y-3x}{c}=1\to \left(2\right)$
Required pair of straight lines can be obtained by homogenisation of curve $\left(1\right)$ using line $\left(2\right)$
$\Rightarrow x^2+y^2=4{\left(\frac{y-3x}{c}\right)}^2\Rightarrow (c^2-36)x^2+(c^2-4)y^2+24xy=0$
$\because$ Lines are perpendicular,
$\Rightarrow c^2-36+c^2-4=0\therefore c^2=20$