Let frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1(a | vedclass

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

Question

(A) Given the ellipse equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with $a > b$.The length of the latus rectum is given by $\frac{2b^{2}}{

Vedclass · Accepted Answer

$126$

Vedclass · Answer

(A) Given the ellipse equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with $a > b$.The length of the latus rectum is given by $\frac{2b^{2}}{a} = 10$,which implies $b^{2} = 5a$ ... $(i)$.Now,consider the function $\phi(t) = \frac{5}{12} + t - t^{2}$.To find the maximum value,we complete the square: $\phi(t) = -\left(t^{2} - t + \frac{1}{4}ight) + \frac{1}{4} + \frac{5}{12} = -\left(t - \frac{1}{2}ight)^{2} + \frac{8}{12} = -\left(t - \frac{1}{2}ight)^{2} + \frac{2}{3}$.The maximum value is $\phi(t)_{	ext{max}} = \frac{2}{3}$,so $e = \frac{2}{3}$.Since $e^{2} = 1 - \frac{b^{2}}{a^{2}}$,we have $\frac{4}{9} = 1 - \frac{b^{2}}{a^{2}}$,which implies $\frac{b^{2}}{a^{2}} = \frac{5}{9}$,so $b^{2} = \frac{5}{9}a^{2}$ ... $(ii)$.Equating $(i)$ and $(ii)$,$5a = \frac{5}{9}a^{2} \Rightarrow a = \frac{a^{2}}{9} \Rightarrow a = 9$.Then $a^{2} = 81$ and $b^{2} = 5(9) = 45$.Therefore,$a^{2} + b^{2} = 81 + 45 = 126$.

Let $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b)$ be a given ellipse whose latus rectum length is $10$. If its eccentricity $e$ is the maximum value of the function $\phi(t) = \frac{5}{12} + t - t^{2}$,then $a^{2} + b^{2}$ is equal to:

Similar Questions

In an ellipse,the distance between its foci is $6$ and the length of the minor axis is $8$. Then its eccentricity is

The eccentricity of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ is:

If the lines $2x - y + 3 = 0$ and $4x + ky + 3 = 0$ are conjugate with respect to the ellipse $5x^2 + 6y^2 - 15 = 0$,then $k$ equals:

An ellipse inscribed in a semi-circle touches the circular arc at two distinct points and also touches the bounding diameter. Its major axis is parallel to the bounding diameter. When the ellipse has the maximum possible area,its eccentricity is

If the distance between the foci of an ellipse is half the length of its latus rectum,then the eccentricity of the ellipse is

Vedclass Test Series

Exam Paper Generator

Online Exam Module