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 with length of its transverse axis $8\sqrt{2}$
• D. an ellipse with length of its major axis $8\sqrt{2}$

Answer 1

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

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