Question

An ellipse has OB as semi-minor axis, F and F’ are its foci and the angle FBF’ is a right angle then, the eccentricity of the ellipse is
$$\begin{gathered} a.{\displaystyle \frac{1}{\sqrt{3}}} \\ b.{\displaystyle \frac{1}{4}} \\ c.{\displaystyle \frac{1}{2}} \\ d.{\displaystyle \frac{1}{\sqrt{2}}} \end{gathered}$$

Answer 1

Hint: In order to solve this problem we need to know that the Product of slopes of two perpendicular lines is -1. Drawing the diagram will help you a lot. You need to use the formula of eccentricity $e=\sqrt{1-{\displaystyle \frac{b^2}{a^2}}}$. Doing this will solve this problem.

Complete step-by-step answer:
The figure to this problem can be drawn as,

Let equation of ellipse is ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$
We can easily see in figure coordinates of B(0, b), F (ae, 0) and F’(-ae, 0).
We know, F’B perpendicular to FB.
Thus, Product of slopes of two perpendicular lines is -1.
So, slope of F’B x slope of FB = -1
Slope of F’B = ${\displaystyle \frac{b-0}{0+ae}}={\displaystyle \frac{b}{ae}}$
Slope of FB = ${\displaystyle \frac{b-0}{0-ae}}={\displaystyle \frac{-b}{ae}}$
$\Rightarrow {\displaystyle \frac{b}{ae}}\times {\displaystyle \frac{-b}{ae}}=-1$ (When we multiply the slopes of two perpendicular lines)
 $\Rightarrow {\displaystyle \frac{-b^2}{a^2{e}^2}}=-1$
On solving we get,
$b^2=a^2{e}^2$…..(1)
We know that $e=\sqrt{1-{\displaystyle \frac{b^2}{a^2}}}$
Now, put the value of $b^2$ from (1) equation in (2) equation.
We get the new equation as,
$$\begin{gathered} \Rightarrow e=\sqrt{1-{\displaystyle \frac{a^2{e}^2}{a^2}}} \\ \Rightarrow e=\sqrt{1-e^2} \end{gathered}$$
On squaring both sides we get,
$\Rightarrow e^2=1-e^2$
On further solving the equations we get,
$$\begin{gathered} \Rightarrow 2e^2=1 \\ \Rightarrow e={\displaystyle \frac{1}{\sqrt{2}}} \end{gathered}$$

So, the correct answer is “Option d”.

Note: Whenever we face such types of problems we use some important points. Like first of all draw a figure and mark coordinates then find the value of slope of lines using coordinates and as we know the product of slopes of two perpendicular lines always be -1. Knowing this will solve our problem and will give you the right answer.