Let an ellipse with centre (1,0) and latus rectum of length frac{1}{2} have its major axis along x -

English

Gujarati

Hindi

10-2. Parabola, Ellipse, Hyperbola

hard

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its minor axis subtends an angle $60^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to $...........$.

A

$9$

B

$8$

C

$7$

D

$6$

(JEE MAIN-2023)

Solution

$\text { L.R. }=\frac{2 b^2}{a}=\frac{1}{2}$

$4 b^2=a…….(i)$

$\text { Ellipse } \frac{(x-1)^2}{a^2}+\frac{y^2}{b^2}=1$

$m_{B_2 F_1}=\frac{1}{\sqrt{3}}$

$\frac{b}{a}=\frac{1}{\sqrt{3}}$

$3 b^2=a^2 e^2=a^2-b^2$

$4 b^2=a^2……..(ii)$

$\text { From (i) and (ii) }$

$a=a^2$

$\therefore a=1$

$b^2=\frac{1}{4}$

$((2 a)+(2 b))^2=9$

Standard 11

Mathematics

Share

Similar Questions

Let $P$ be a variable point on the ellipse $x^2 + 3y^2 = 3$ , then the maximum perpendicular distance of $P$ from the line $x -y = 10$ is

normal

The equation of an ellipse, whose vertices are $(2, -2), (2, 4)$ and eccentricity $\frac{1}{3}$, is

hard

Consider two straight lines, each of which is tangent to both the circle $x ^2+ y ^2=\frac{1}{2}$ and the parabola $y^2=4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O (0,0)$ and whose semi-major axis is $OQ$. If the length of the minor axis of this ellipse is $\sqrt{2}$, then which of the following statement($s$) is (are) $TRUE$?

$(A)$ For the ellipse, the eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $1$

$(B)$ For the ellipse, the eccentricity is $\frac{1}{2}$ and the length of the latus rectum is $\frac{1}{2}$

$(C)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{4 \sqrt{2}}(\pi-2)$

$(D)$ The area of the region bounded by the ellipse between the lines $x=\frac{1}{\sqrt{2}}$ and $x=1$ is $\frac{1}{16}(\pi-2)$

normal

(IIT-2018)

If a number of ellipse be described having the same major axis $2a$ but a variable minor axis then the tangents at the ends of their latera recta pass through fixed points which can be

normal

An ellipse has $OB$ as semi minor axis, $F$ and $F'$ its foci and the angle $FBF'$ is a right angle. Then the eccentricity of the ellipse is

hard

(AIEEE-2005)