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), is

• A. an ellipse whose eccentricity is $\frac{1}{\sqrt{3}}$
• B. a hyperbola whose eccentricity is $\sqrt{3}$
• C. a hyperbola having length of its transverse axis $8\sqrt{2}$ units
• D. an ellipse having length of its major axis $8\sqrt{2}$ units

Answer 1

The correct option is C a hyperbola having length of its transverse axis $8\sqrt{2}$ units

$\sqrt{2}x-y=-4\sqrt{2}k.....\left(1\right)$
$\sqrt{2}x+y=\frac{4\sqrt{2}}{k}.....\left(2\right)$
Multiplying both equation
$(\sqrt{2}x+y)(\frac{\sqrt{2}x-y}{-4\sqrt{2}})=4\sqrt{2}$
$\Rightarrow 2x^2-y^2=-32$
$\Rightarrow \frac{y^2}{32}-\frac{x^2}{16}=1$
Hence, it represents a hyperbola having transverse axis along $y-$axis
$\Rightarrow e=\sqrt{1+\frac{16}{32}}=\sqrt{\frac{3}{2}}$
Also, length of transverse axis $2\sqrt{32}=8\sqrt{2}$ units