If the foci of a hyperbola are the same as th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) For the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$,we have $a^2 = 25$ and $b^2 = 9$.The eccentricity $e_e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1

Vedclass · Accepted Answer

$\frac{x^2}{4} - \frac{y^2}{12} = 1$

Vedclass · Answer

(A) For the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$,we have $a^2 = 25$ and $b^2 = 9$.The eccentricity $e_e = \sqrt{1 - \frac{b^2}{a^2}} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$.The foci are $(\pm a_e e_e, 0) = (\pm 5 	imes \frac{4}{5}, 0) = (\pm 4, 0)$.For the hyperbola,the foci are $(\pm 4, 0)$,so $ae = 4$. Given $e = 2$,we have $a = \frac{4}{2} = 2$.For a hyperbola,$b^2 = a^2(e^2 - 1) = 2^2(2^2 - 1) = 4(4 - 1) = 4(3) = 12$.Thus,the equation of the hyperbola is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,which is $\frac{x^2}{4} - \frac{y^2}{12} = 1$.

If the foci of a hyperbola are the same as the foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ and the eccentricity of the hyperbola is $2$,then what is its equation?

Similar Questions

$(p, q)$ is the point of intersection of a latus rectum and an asymptote of the hyperbola $9x^2 - 16y^2 = 144$. If $p > 0$ and $q > 0$,then $q =$

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $6$. If $(\sqrt{13}, k)$ is an endpoint of a latus rectum of this hyperbola,then $k=$

Let $C$ be the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $P$ be a point on it. If the tangent at $P$ to the hyperbola meets the straight lines $bx-ay=0$ and $bx+ay=0$ respectively in $Q$ and $R$,then $CQ \cdot CR=$

If the normal to the rectangular hyperbola $x^2-y^2=1$ at the point $P$ with parameter $\theta_1 = \frac{\pi}{4}$ meets the curve again at $Q$ with parameter $\theta_2$,then find the value of $\sec^2 \theta_2 + \tan \theta_2$.

If a circle intersects the hyperbola $xy = 1$ at four points $(x_r, y_r)$ for $r = 1, 2, 3, 4$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module