Question

A focal chord of the parabola $y^2=8x$ is tangent to the circle $(x-5{)}^2+y^2=3$, then the possible value of the twice of square of slope of this chord is

Answer 1

Focus of $y^2=8x$ is (2, 0).
Any focal chord is $y-0=m(x-2)$
or $mx-y-2m=0$
This focal chord touches the circle $(x-5{)}^2+y^2=3$
Then distance from the centre of this circle to the chord is equal to radius of the circle
or $\frac{|5m-2m|}{\sqrt{m^2+1}}=\sqrt{3}$
$\Rightarrow 3m=\sqrt{3}.\sqrt{m^2+1}$
$\Rightarrow 9m^2=3m^2+3\Rightarrow 2m^2=1$