In an ellipse,its foci and the ends of its ma | vedclass

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

Question

(D) Let the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,where $a > b$.The ends of the major axis are $A'(-a, 0)$ and $A(a, 0)$.The foci are $S'

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(D) Let the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,where $a > b$.The ends of the major axis are $A'(-a, 0)$ and $A(a, 0)$.The foci are $S'(-ae, 0)$ and $S(ae, 0)$.Given that the foci and the ends of the major axis are equally spaced,we have the segments $A'S'$,$S'S$,and $SA$ equal in length.$A'S' = S'S = SA = k$ (say).The total length of the major axis is $A'A = 2a$.Thus,$k + k + k = 2a \implies 3k = 2a \implies k = \frac{2a}{3}$.The distance between the foci is $S'S = 2ae = k = \frac{2a}{3}$.Therefore,$e = \frac{1}{3}$.We know the relation $b^2 = a^2(1 - e^2)$.Given $b = 2\sqrt{2}$,so $b^2 = 8$.$8 = a^2(1 - (\frac{1}{3})^2) = a^2(1 - \frac{1}{9}) = a^2(\frac{8}{9})$.$8 = \frac{8a^2}{9} \implies a^2 = 9 \implies a = 3$.The length of the semi-major axis is $3$.

In an ellipse,its foci and the ends of its major axis are equally spaced. If the length of its semi-minor axis is $2 \sqrt{2}$,then the length of its semi-major axis is

Similar Questions

If $4x - 3y - 5 = 0$ is a normal to the ellipse $3x^2 + 8y^2 = k$,then the equation of the tangent drawn to this ellipse at the point $(-2, m)$ $(m > 0)$ is

Let $P$ be any point on the ellipse $7x^2 + 16y^2 = 112$,$S$ be a focus,$L$ be the corresponding directrix,and $PM$ be the perpendicular distance from $P$ to the directrix $L$. Then $\frac{SP}{PM} =$

The circumcenter of the equilateral triangle having the three points $\theta_1, \theta_2, \theta_3$ lying on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ as its vertices is $(r, s)$. Then the average of $\cos(\theta_1-\theta_2)$,$\cos(\theta_2-\theta_3)$ and $\cos(\theta_3-\theta_1)$ is

The equation of the ellipse whose distance between the foci is $8$ and the distance between the directrices is $18$,is

The lengths of the major and minor axes of an ellipse are $10$ and $8$ respectively,and its major axis lies along the $y$-axis. The equation of the ellipse,with its center at the origin,is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module