# Ellipse (S.N.Dey)|Part-1|Ex-5

##### In this article, we will do complete solutions of Very Short Answer Type Questions of Ellipse Chapter of  S.N.Dey mathematics, Class 11.

Find the length of the latus rectum and the co-ordinate of the foci of the ellipse

Solution.

The given equation of ellipse can be written as

Comparing with we get

The length of the latus rectum is

Now, eccentricity of the ellipse

The co-ordinates of foci are

Find the length of the latus rectum of the ellipse

Solution.

We have, the given equation of ellipse

Comparing with we get,

So, the length of the latus rectum is given by

Calculate the eccentricity of the ellipse

Solution.

Comparing the given ellipse with we get,

So, the eccentricity of the ellipse is given by

Find the equations of the directrices of the ellipse

Solution.

The given equation of the ellipse can be written as

Comparing with we get,

So, the equations of the directrices of the given ellipse are

Find the distance between the foci of the ellipse

Solution.

The equation of the ellipse can be written as

Comparing with we get,

Now, the eccentricity of the ellipse is

So, the distance between the foci

Find the eccentricity of the ellipse if

the length of the latus rectum is equal to half the minor axis of the ellipse.

Solution.

Let the equation of the ellipse be

Now, the length of the latus rectum and the length of the half the minor axis unit.

So, by condition,

the length of the minor axis is equal to half the distance between the foci of the ellipse.

Solution.

Given :

Minor axis= (Distance between the foci)

the length of minor axis is equal to the distance between the later recta.

Solution.

We know that the distance between the foci of the ellipse where

By question,

So, by and we get,

If the ellipses and have same eccentricity, show that

We have the equations of two ellipses which are

The eccentricity of

The eccentricity of

The ellipse has the same eccentricity as the ellipse  . Find the ratio

Solution.

We have the given equation of ellipse

The eccentricity of

Again, the eccentricity of is

Find the positions of the points and with respect to the ellipse

Solution.

The equation of the ellipse can be written as

Case-1 :

So, the point lies inside the ellipse.

Case-2:

So, the point lies on the ellipse.

Case-3:

So, the point lies outside the ellipse.

For what values of does the point lie outside the ellipse

The co-ordinates of a point on the ellipse are ; find the eccentric angle of the point.

Solution.

The equation of the given ellipse can be written in standard form as

Any point on can be taken as

Now, by question, we can have

Hence, the eccentric angle of the point is

Find the co-ordinates of the point on the ellipse whose eccentric angle is

Solution.

The equation of the ellipse can be written as

Comparing with the general equation of the ellipse we get,

So, any point on the given ellipse with the eccentric angle can be written as

So, the co-ordinates of the point on the ellipse whose eccentric angle is is given by

If the ellipse prove that

Solution.

We have the equations of two ellipses which are

The eccentricity of

The eccentricity of