SATHEE: Ellipse 1 Question 8

Ellipse 1 Question 8

9. An ellipse has $O B$ as a semi-minor axis. $F$ and $F^{'}$ are its foci and the angle $F B F^{'}$ 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 $F B F^{'}$ is right angled.

$∴$ (Slope of $F B) \cdot ($ Slope of $F^{'} B) = - 1$

$\begin{aligned} \Rightarrow & \frac{0 - b}{a e - 0} \cdot \frac{0 - b}{- a e - 0} & = - 1 \\ \Rightarrow & \frac{b^{2}}{- a^{2} e^{2}} & = - 1 \Rightarrow b^{2} = a^{2} e^{2} \\ \Rightarrow & a^{2} (1 - e^{2}) & = a^{2} e^{2} \\ \Rightarrow & e^{2} & = 1 / 2 \Rightarrow e = 1 / \sqrt{2} \end{aligned}$

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Ellipse 1 Question 8

9. An ellipse has OB as a semi-minor axis. F and F′ are its foci and the angle FBF′ is a right angle. Then, the eccentricity of the ellipse is …… .

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9. An ellipse has $O B$ as a semi-minor axis. $F$ and $F^{'}$ are its foci and the angle $F B F^{'}$ is a right angle. Then, the eccentricity of the ellipse is …… .