The locus of the point of intersection of the lines 2 x - y + 4 2 k...

Given equation of lines are $\sqrt{2}x - y + 4\sqrt{2}k = 0 . . . \left(1\right)<p>and$\sqrt{2}kx + ky - 4\sqrt{2} = 0 . . . \left(2\right)$</p><p>Eliminating$k$from \left(1\right) and \left(2\right),</p><p>$⇒ \left(\sqrt{2}x + y\right)\left(\dfrac{\sqrt{2}x - y}{- 4\sqrt{2}}\right) = 4\sqrt{2}$</p><p>$⇒ 2{x}^{2} - {y}^{2} = - 32$</p><p>$⇒ \dfrac{{y}^{2}}{32} - \dfrac{{x}^{2}}{16} = 1$, which is a hyperbola.</p><p>$⇒ e = \sqrt{1 + \dfrac{16}{32}} = \sqrt{\dfrac{3}{2}} and length of transverse axis

$= 8\sqrt{2}$.