Locus of point of intersection of lines (√{3}) kx + ky -4 √{3}=0 and √{3} x-y-4(√{3}) k=0 is

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

The locus of the point of intersection of the lines $(\sqrt{3}) kx + ky -4 \sqrt{3}=0$ and $\sqrt{3} x-y-4(\sqrt{3}) k=0$ is a conic, whose eccentricity is .............

A

$0$

B

$2$

C

$4$

D

$8$

(JEE MAIN-2021)

Solution

$K =\frac{4 \sqrt{3}}{\sqrt{3} x + y }=\frac{\sqrt{3} x – y }{4 \sqrt{3}}$

$\Rightarrow 3 x ^{2}- y ^{2}=48$

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

Now, $48=16\left( e ^{2}-1\right)$

$\Rightarrow e =\sqrt{4}=2$

Standard 11

Mathematics

Share

Similar Questions

The tangent to the hyperbola, $x^2 – 3y^2 = 3$ at the point $\left( {\sqrt 3 \,\,,\,\,0} \right)$ when associated with two asymptotes constitutes :

normal

Point from which two distinct tangents can be drawn on two different branches of the hyperbola $\frac{{{x^2}}}{{25}} – \frac{{{y^2}}}{{16}} = \,1$ but no two different tangent can be drawn to the circle $x^2 + y^2 = 36$ is

normal

The length of the latus rectum of the hyperbola $25x^2 -16y^2 = 400$ is –

normal

The eccentricity of the hyperbola can never be equal to

easy

Let the focal chord of the parabola $P: y^{2}=4 x$ along the line $L: y=m x+c, m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H : x ^{2}- y ^{2}=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $OMFN$ is.

normal

(JEE MAIN-2022)