If the end points of the latus rectum of an e | 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 coordinates of the end points of the latus rectum are $(ae, \frac{b^2

Vedclass · Accepted Answer

$\frac{\sqrt{5} - 1}{2}$

Vedclass · Answer

(D) Let the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$.The coordinates of the end points of the latus rectum are $(ae, \frac{b^2}{a})$ and $(ae, -\frac{b^2}{a})$.Since these points are vertices of a square,the distance between them must be equal to the distance between the two end points of the latus rectum on the same side of the major axis,which is $2ae$.Thus,$2 	imes \frac{b^2}{a} = 2ae$.$\frac{b^2}{a} = ae$.Using $b^2 = a^2(1 - e^2)$,we get $\frac{a^2(1 - e^2)}{a} = ae$.$a(1 - e^2) = ae \Rightarrow 1 - e^2 = e$.$e^2 + e - 1 = 0$.Solving for $e$ using the quadratic formula,$e = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{5}}{2}$.Since $e > 0$,we have $e = \frac{\sqrt{5} - 1}{2}$.

If the end points of the latus rectum of an ellipse are the vertices of a square,then the eccentricity of the ellipse will be -

Similar Questions

The distance between the directrices of the ellipse $\frac{x^2}{36}+\frac{y^2}{20}=1$ is

If $P$ is any point on the ellipse $16x^2 + 25y^2 = 400$ with foci $S$ and $S'$, and the area of $\Delta PSS'$ is $9 \text{ square units}$, then the abscissa of point $P$ is:

If the coordinates of two points $A$ and $B$ are $(\sqrt{7}, 0)$ and $(-\sqrt{7}, 0)$ respectively and $P$ is any point on the conic $9x^{2} + 16y^{2} = 144$,then $PA + PB$ is equal to

The equation of the locus of the foot of the perpendicular drawn from the centre of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ to any tangent of the ellipse is

An ellipse has $6$ and $2$ as the lengths of its major and minor axes,respectively. If the center is at $(5,6)$ and the major axis is along $x-y+1=0$,then the equation of the ellipse is

Vedclass Test Series

Exam Paper Generator

Online Exam Module