Question

The locus of the point of intersection of the lines $\sqrt{3}x-y-4\sqrt{3k}=0$ and $\sqrt{3k}x+ky-4\sqrt{3}=0$ for different values of $k$ is a

• A. ellipse
• B. parabola
• C. circle
• D. hyperbola

Answer 1

Answer: (D)

hyperbola

Let point of intersection is $(x_1,y_1)$

$So,\sqrt{3}{x}_1-y_1=4\sqrt{3}k$ ....(i)

$\sqrt{3}k{x}_1+k{y}_1=4\sqrt{3}$ ....(ii)

Multiply (i) and (ii) we get $3k{x}_1^2-k{y}_1^2=48$.

Therefore, the required locus is, $3x^2-y^2=48$, which is a hyperbola.
Hence, option 'D' is correct.