Mathematics - Point Question with Solution | TestHub

Given equation of lines are$\sqrt{2}x-y+4\sqrt{2}k=0 ...\left(1\right)$

and $\sqrt{2}kx+ky-4\sqrt{2}=0 ...\left(2\right)$ 

Eliminating $k$ from $\left(1\right)$ and $\left(2\right)$,

$\Rightarrow \left(\sqrt{2}x+y\right)\left(\frac{\sqrt{2}x-y}{-4\sqrt{2}}\right)=4\sqrt{2}$

$\Rightarrow 2x^2-y^2=-32$

$\Rightarrow \frac{y^2}{32}-\frac{x^2}{16}=1$, which is a hyperbola.

$\Rightarrow e=\sqrt{1+\frac{16}{32}}=\sqrt{\frac{3}{2}}$ and length of transverse axis

$=8\sqrt{2}$.