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# 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.

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

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

• 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$
$3x^2-y^2=48$
$\Rightarrow \large\frac{x^2}{16}-\frac{y^2}{48}$$=1$