Ellipse 1 Question 8
9. An ellipse has $O B$ as a semi-minor axis. $F$ and $F^{\prime}$ are its foci and the angle $F B F^{\prime}$ is a right angle. Then, the eccentricity of the ellipse is …… .
$(1997,2 M)$
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Answer:
Correct Answer: 9. $e=1 / \sqrt{2}$
Solution:
- Since, angle $F B F^{\prime}$ is right angled.
$\therefore \quad$ (Slope of $F B) \cdot\left(\right.$ Slope of $\left.F^{\prime} B\right)=-1$
$$ \begin{aligned} & \Rightarrow & \frac{0-b}{a e-0} \cdot \frac{0-b}{-a e-0} & =-1 \\ & \Rightarrow & \frac{b^{2}}{-a^{2} e^{2}} & =-1 \Rightarrow b^{2}=a^{2} e^{2} \\ \Rightarrow & & a^{2}\left(1-e^{2}\right) & =a^{2} e^{2} \\ & \Rightarrow & e^{2} & =1 / 2 \Rightarrow e=1 / \sqrt{2} \end{aligned} $$
