Exam

BITSAT

Year

2013

Subject

Mathematics

Chapter

Ellipse

BITSAT 2013 · Mathematics · Ellipse

An ellipse has O B as semi minor axis, F and F^ its focii and the...

An ellipse has $O B$ as semi minor axis, $F$ and $F^{\prime}$ its focii and the angle $F B F^{\prime}$ is a right angle. Then the eccentricity of the ellipse is

Step-by-step solution

$$\begin{aligned} & \because \angle F B F^{\prime}=90^{{}^{\circ}} \Rightarrow F B^{2}+F^{\prime} B^{2}=F F^{\prime 2} \\ \therefore &\left(\sqrt{a^{2} e^{2}+b^{2}}\right)^{2}+\left(\sqrt{a^{2} e^{2}+b^{2}}\right)^{2}=(2 a e)^{2} \\ & \Rightarrow 2\left(a^{2} e^{2}+b^{2}\right)=4 a^{2} e^{2} \Rightarrow e^{2}=\frac{b^{2}}{a^{2}} \ldots(1) \end{aligned}$$
Also, $e^{2}=1-b^{2} / a^{2}=1-e^{2}$ (By using equation (i)) $\Rightarrow 2 e^{2}=1 \Rightarrow e=\frac{1}{\sqrt{2}}$