Let the length of the latus rectum of an elli | vedclass

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

Question

(C) The function is $f(t) = -t^{2} + 2t - \frac{3}{4}$.To find the maximum value, we complete the square: $f(t) = -(t^{2} - 2t + 1) + 1 - \frac{3}{4}

Vedclass · Accepted Answer

$496$

Vedclass · Answer

(C) The function is $f(t) = -t^{2} + 2t - \frac{3}{4}$.To find the maximum value, we complete the square: $f(t) = -(t^{2} - 2t + 1) + 1 - \frac{3}{4} = -(t-1)^{2} + \frac{1}{4}$.The maximum value is $e = \frac{1}{4}$, so $e^{2} = \frac{1}{16}$.For an ellipse, $e^{2} = 1 - \frac{b^{2}}{a^{2}}$, so $\frac{b^{2}}{a^{2}} = 1 - \frac{1}{16} = \frac{15}{16} \Rightarrow b^{2} = \frac{15}{16}a^{2}$.The length of the latus rectum is $\frac{2b^{2}}{a} = 30$, so $b^{2} = 15a$.Equating the expressions for $b^{2}$: $\frac{15}{16}a^{2} = 15a$.Since $a 
eq 0$, we have $\frac{a}{16} = 1 \Rightarrow a = 16$.Then $b^{2} = 15(16) = 240$.Thus, $a^{2} + b^{2} = 16^{2} + 240 = 256 + 240 = 496$.

Let the length of the latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b)$ be $30$. If its eccentricity is the maximum value of the function $f(t)=-\frac{3}{4}+2t-t^{2}$, then $(a^{2}+b^{2})$ is equal to -

Similar Questions

Consider the conic $e x^2 + \pi y^2 - 2 e^2 x - 2 \pi^2 y + e^3 + \pi^3 = \pi e$.
Suppose $P$ is any point on the conic and $S_1, S_2$ are the foci of the conic,then the maximum value of $(P S_1 + P S_2)$ is

Find the coordinates of the foci,the vertices,the lengths of the major and minor axes,and the eccentricity of the ellipse $9x^{2} + 4y^{2} = 36$.

The foci of the conic $25x^2 + 16y^2 - 150x = 175$ are

The equation of the tangent to the curve $9x^{2} + 16y^{2} = 144$ which makes equal intercepts with the coordinate axes is:

An ellipse has its major axis along the $y$-axis and its minor axis along the $x$-axis. If the length of its latus rectum is $\frac{2}{3}$ times the length of its minor axis,then the eccentricity of the ellipse is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module