# Ellipse (S.N.Dey) | Part-4 | Ex-5

###### In the previous article , we solved few solutions of Short Answer Type Questions of Ellipse Chapter of  S.N.Dey mathematics, Class 11. In this chapter, we will solve few more.

A point moves so that its distance from is times its distance from the line Show that the locus of the moving point is an ellipse whose equation you are to determine.

Solution.

Let the co-ordinates of any point be so that distance of the point from the point is

Again, distance of the point from the line is

By question,

Hence, the equation of the moving point is an ellipse and is given by

Find the equation of the ellipse whose major axis is parallel to axis and

centre is , eccentricity is and the length of latus rectum is

centre is length of major axis and the co-ordinates of foci are and

Solution (i)

The length of the latus rectum

Hence, the equation of the ellipse is given by

Solution(ii)

The length of the major axis

The distance between the foci is

Hence, the equation of the ellipse is

Find the equation of the ellipse , for which the foci are and and length of the minor axis is unit. Explain how the ellipse is reduced to a circle when its two foci coincide.

Solution.

The length of minor axis

The distance between two foci is given by

From the position of foci we can easily conclude that the ellipse will be of the form

2nd Part :

When two foci coincide, then the distance between two foci

Hence, the equation of the ellipse is reduced into

So, the equation represents a circle.

The eccentricity of an ellipse is focus is and the major axis and directrix intersect at Find the co-ordinates of the centre of the ellipse.

Solution.

Let the centre of the ellipse be Now, the focus divides the line segment at the ratio internally.

Hence, the centre of the ellipse is

The lengths of major and minor axes of an ellipse are and and their equations are and respectively. Find the equation of the ellipse.

Solution.

Since the major axis and minor axes are parallel to the -axis and -axis respectively. So, the equation of the ellipse can be taken in the form where is the centre of the ellipse.

The length of the major axis :

The length of the major axis :

Now, the centre of the ellipse is the intersection of and which is

Hence, the equation of the ellipse is

Show that the point lies on the ellipse Show further that the sum of its distances from two foci is equal to the length of its major axis.

Solution.

Since the point satisfies the equation of the ellipse, the point lies on the ellipse.

Again,

So,

Hence, the co-ordinates of the foci is

So, the sum of its distances from the two foci is

Prove that for the ellipse and are two foci of the ellipse of the ellipse and is any point on the ellipse.

Solution.

The equation of the given ellipse is i.e.,

Comparing with , we get

So, the co-ordinates of any point on the ellipse can be written as

So, the co-ordinates of the foci