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

Answer 1

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

Given lines are :

$\sqrt{2}x-y+4\sqrt{2}k=0$

$\Rightarrow \sqrt{2}x+4\sqrt{2}k=y$ ..... $\left(i\right)$ and

$\sqrt{2}kx+ky-4\sqrt{2}=0$ ..... $\left(ii\right)$

We have from the equations of the lines:

Substituting $\left(i\right)$ in $\left(ii\right)$,

$\Rightarrow 2\sqrt{2}kx+4\sqrt{2}(k^2-1)=0$

$\Rightarrow x=\frac{2(1-k^2)}{k},y=\frac{2\sqrt{2}(1+k^2)}{k}$

$\Rightarrow {\left(\frac{y}{4\sqrt{2}}\right)}^2-{\left(\frac{x}{4}\right)}^2=1$

$\Rightarrow {\left(\frac{y}{4\sqrt{2}}\right)}^2-{\left(\frac{x}{4}\right)}^2=1$

Locus of transverse axis

$=2\sqrt{32}$

$=2\times 4\sqrt{2}$

$=8\sqrt{2}$

Thus, the locus is a hyperbola with length of its transverse axis equal to $8\sqrt{2}$.

So option A is the correct answer.