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. an ellipse with length of its major axis $8\sqrt{2}$
• C. a hyperbola whose eccentricity is $\sqrt{3}$
• D. a hyperbola with length of its transverse axis $8\sqrt{2}$

Answer 1

Answer: (D)

Given:
$\sqrt{2}x-y+4\sqrt{2}k=0$
$\sqrt{2}kx+ky-4\sqrt{2}=0$
Now, eliminating $k$ from Eq. (2) by putting value of $k$ from Eq. (1), we get
$(\sqrt{2}x+y)\left(\frac{\sqrt{2x}-y}{-4\sqrt{2}}\right)=4\sqrt{2}$
$2x^2-y^2=-32$
$\frac{y^2}{32}-\frac{x^2}{16}=1$
The above equation represents the hyperbola.
So, eccentricity e of this hyperbola is $e=\sqrt{1+\frac{16}{32}}=\sqrt{\frac{3}{2}}$
Length of transverse axis $=8\sqrt{2}$