State whether the following statement is True or False : The locus of the point of intersection of lines $\sqrt 3x-y-4\sqrt 3k=0$ and $\sqrt 3kx+ky-4\sqrt 3=0$ for different value of $k$ is a hyperbola whose eccentricity is 2.

Question

[Hint : Eliminate $k$ between the given equations]

$\begin{array}{1 1}\text{True}\\\text{False}\end{array} $

Answer 1

Toolbox:

• General equation of a hyperbola whose foci lie on $x$-axis is $\large\frac{x^2}{a^2}-\frac{y^2}{b^2}$$=1$

Answer : True

The given equations are $\sqrt 3x-y-4\sqrt 3k=0$-----(1) and $\sqrt 3kx+ky-4\sqrt 3=0$----(2)

From equ(1) we get

$k=\large\frac{\sqrt 3x-y}{4\sqrt 3}$

Substituting the value of $'k'$ in equ(2) we get,

$\sqrt 3x\big(\large\frac{\sqrt 3x-y}{4\sqrt 3}\big)+\big(\large\frac{\sqrt 3x-y}{4\sqrt 3}\big)$$y=4\sqrt 3$

On simplifying we get,

$3x^2-y^2=48$

$\Rightarrow \large\frac{x^2}{16}-\frac{y^2}{48}$$=1$

This is clearly an equation of a hyperbola.

Hence the statement is True.