Answer

Verified

429k+ views

**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 - \dfrac{{{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 $\dfrac{{{x^2}}}{{{a^2}}} + \dfrac{{{y^2}}}{{{b^2}}} = 1$

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 = $\dfrac{{b - 0}}{{0 + ae}} = \dfrac{b}{{ae}}$

Slope of FB = $\dfrac{{b - 0}}{{0 - ae}} = \dfrac{{ - b}}{{ae}}$

$ \Rightarrow \dfrac{b}{{ae}} \times \dfrac{{ - b}}{{ae}} = - 1$ (When we multiply the slopes of two perpendicular lines)

$ \Rightarrow \dfrac{{ - {b^2}}}{{{a^2}{e^2}}} = - 1$

On solving we get,

${b^2} = {a^2}{e^2}$…..(1)

We know that $e = \sqrt {1 - \dfrac{{{b^2}}}{{{a^2}}}} $

Now, put the value of ${b^2}$ from (1) equation in (2) equation.

We get the new equation as,

$

\Rightarrow e = \sqrt {1 - \dfrac{{{a^2}{e^2}}}{{{a^2}}}} \\

\Rightarrow e = \sqrt {1 - {e^2}} \\

$

On squaring both sides we get,

$ \Rightarrow {e^2} = 1 - {e^2}$

On further solving the equations we get,

$

\Rightarrow 2{e^2} = 1 \\

\Rightarrow e = \dfrac{1}{{\sqrt 2 }} \\

$

**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.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

10 examples of evaporation in daily life with explanations