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

$$\begin{gathered} \mathrm{Given} \mathrm{lines} \mathrm{are} \sqrt{2}\mathrm{x}-\mathrm{y}=-4\sqrt{2}\mathrm{k} — ① \\ \mathrm{k}\left(\sqrt{2}\mathrm{x}+\mathrm{y}\right)=4\sqrt{2} \Rightarrow \mathrm{k}=\frac{4\sqrt{2}}{\sqrt{2} \mathrm{x}+\mathrm{y}} — ② \\ \mathrm{from} ① \& ② \sqrt{2}\mathrm{x}-\mathrm{y}=-4\sqrt{2}\left[\frac{4\sqrt{2}}{\sqrt{2}\mathrm{x}+\mathrm{y}}\right] \\ \Rightarrow \left(\sqrt{2}\mathrm{x}-\mathrm{y}\right) \left(\sqrt{2}\mathrm{x}+\mathrm{y}\right)=-32 \\ \Rightarrow 2{\mathrm{x}}^2-{\mathrm{y}}^2=-32 \\ \Rightarrow \frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{32}=-1 \\ \mathrm{length} \mathrm{of} \mathrm{transverse} \mathrm{axis} =2\sqrt{32} \left(\because 2\mathrm{b}\right) \\ =2\times 4\sqrt{2} \\ =8\sqrt{2} \end{gathered}$$