Let $C_{1}$ and $C_{2}$ represents standard ellipse and hyperbola (centered at origin and x axis as major axis/transverse axis).
Let the foci of $C_{2}$ be the vertices of $C_{1}$ and the foci of $C_{1}$ be the vertices of $C_{2}$ .Let the eccentricity of the ellipse and the hyperbola be $e_{1}$ and $e_{2} .$ The angle between the asymptotes of the hyperbola is $\frac{π}{3} .$ If $e_{1}^{2} + e_{2}^{2} = \frac{p}{q}$ (where $p$ and $q$ are coprime), then the value of $p + q$ is

Open in App

Solution

Let $C_{1} : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $C_{2} : \frac{x^{2}}{A^{2}} - \frac{y^{2}}{B^{2}} = 1$

Clearly, $A = a e_{1} & A e_{2} = a \Rightarrow e_{1} e_{2} = 1$
Now, angle between the asymptotes is $\frac{π}{3}$
$\Rightarrow 2 {tan}^{- 1} \frac{B}{A} = \frac{π}{3} \Rightarrow {tan}^{- 1} \frac{B}{A} = \frac{π}{6} \Rightarrow \frac{B}{A} = \frac{1}{\sqrt{3}} \Rightarrow \frac{B^{2}}{A^{2}} = \frac{1}{3} e_{2}^{2} = 1 + \frac{B^{2}}{A^{2}} = 1 + \frac{1}{3} = \frac{4}{3} \Rightarrow e_{1}^{2} = \frac{3}{4} e_{1}^{2} + e_{2}^{2} = \frac{25}{12} \Rightarrow p + q = 37$

Suggest Corrections

Similar questions

Q. Let an ellipse and a hyperbola have same foci. If the length of conjugate axis of the hyperbola is equal to the length of minor axis of the ellipse, then the value of

\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}

(e_{1}

and

e_{2}

denote the eccentricities of the two conics

)

Q. Let an ellipse and a hyperbola have same foci. If the length of conjugate axis of the hyperbola is equal to the length of minor axis of the ellipse, then the value of

\frac{1}{e_{1}^{2}} + \frac{1}{e_{2}^{2}}

(e_{1}

and

e_{2}

denote the eccentricities of the two conics

)

Q. An ellipse and hyperbola share common foci

F_{1}

and

F_{2}

. The ellipse has vertices at the origin and

(0, 24)

and a semi-minor axis of length

11

. The hyperbola has a conjugate axis of length

4 \sqrt{3}

. The ellipse and hyperbola intersect at four points. Let P be one of the points of intersection, then value of

(P F_{1})^{2} + (P F_{2})^{2}

is?

Let the ellipse $C_{1} : \frac{x^{2}}{a_{1}^{2}} + \frac{y^{2}}{b_{1}^{2}} = 1 (a_{1} > b_{1})$ and the hyperbola $C_{2} : \frac{x^{2}}{a_{2}^{2}} - \frac{y^{2}}{b_{2}^{2}} = 1$ have the same focus point $F_{1}$ and $F_{2}$ . If point $P$ is the intersection point of $C_{1}$ and $C_{2}$ in the first quadrant and $| F_{1} F_{2} | = 2 | P F_{2} |$ , then which of the following is (are) CORRECT?
( $e_{1}$ and $e_{2}$ are eccentricities of ellipse and hyperbola respectively.)