Question

The locus of the point of intersection of the lines, $\sqrt{2}x-y+4\sqrt{2}k=0$ and $\sqrt{2}kx+ky-4\sqrt{2}=0$ ($k$ is any non-zero real parameter) ?

Answer 1

$(\sqrt{2}x+y)R-4\sqrt{2}=0$

$R=\frac{4\sqrt{2}}{\sqrt{2}x+y}$

$\sqrt{2}x-y+4\sqrt{2}\left(\frac{4\sqrt{2}}{\sqrt{2}x+y}\right)=0$

$(\sqrt{2}x-y)(\sqrt{2}x+y)+(4\sqrt{2}{)}^2=0$

$2x^2-y^2+32=0$

$y^2-2x^2=32\Rightarrow \frac{y^2}{32}-\frac{x^2}{16}=1$

Hypaerbola