If angle between lines joining end points of minor axis of an ellipse with its foci is πover2 , the

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

If the angle between the lines joining the end points of minor axis of an ellipse with its foci is $\pi\over2$, then the eccentricity of the ellipse is

A

$1\over2$

B

$1/\sqrt 2 $

C

$\sqrt 3 /2$

D

$1/2\sqrt 2 $

(IIT-1997)

Solution

(b) Since $\angle \,FBF' = \frac{\pi }{2}$ (Given)

$\therefore $$\angle \,FBC = \,\angle \,F'BC = \pi /4$

$CB = CF \Rightarrow \,\,b = ae$

==> ${b^2} = {a^2}{e^2}$

==> ${a^2}(1 – {e^2}) = {a^2}{e^2}$

==> $1 – {e^2} = {e^2}$

==> $2{e^2} = 1$

==> $e = 1/\sqrt 2 $.

Standard 11

Mathematics

Share

Similar Questions

Let the ellipse, $E _1: \frac{ x ^2}{ a ^2}+\frac{ y ^2}{b^2}=1, a > b$ and $E _2: \frac{ x ^2}{A^2}+\frac{ y ^2}{B^2}=1, A< B$ have same eccentricity $\frac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectums be $\frac{32}{\sqrt{3}}$, and the distance between the foci of $E_1$ be $4$. If $E_1$ and $E_2$ meet at $A, B, C$ and $D$, then the area of the quadrilateral $A B C D$ equals:

hard

(JEE MAIN-2025)

The line $x\cos \alpha + y\sin \alpha = p$ will be a tangent to the conic $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, if

easy

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

medium

(JEE MAIN-2025)

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2,3)$

normal

Let $\mathrm{A}(\alpha, 0)$ and $\mathrm{B}(0, \beta)$ be the points on the line $5 x+7 y=50$. Let the point $P$ divide the line segment $A B$ internally in the ratio $7: 3$. Let $3 x-$ $25=0$ be a directrix of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the $\mathrm{x}$-axis passes through $\mathrm{P}$, then the length of the latus rectum of $\mathrm{E}$ is equal to

hard

(JEE MAIN-2024)