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

###### In the previous article , we solved 7 solutions of Short Answer Type Questions (from 8-14) of Ellipse Chapter of  S.N.Dey mathematics, Class 11. In this chapter, we will solve few more.

15. If be a variable parameter, show that the point always lies on an ellipse.

Solution.

From and we get,

Clearly, equation represents an ellipse. Hence, the point always lies on an ellipse.

16. A point moves on a plane in such a manner that the sum of its distances from the points and is always constant and equal to Show that the locus of the moving point is the ellipse

Solution.

Let the co-ordinates of the moving point be Also, let

Hence, by we can conclude that the locus of moving point which represents an ellipse.

17. Find the locus of the point , the ratio of whose distances from the line and from the point is

Solution.

Let the co-ordinates of the point be

The distance of P from the line is

Again, the distance of P from the point is

.

By question,

Hence, the locus of the point is

18. The lengths of the major and minor axes of an ellipse are and and is the foot of the perpendicular drawn from a point of the ellipse on the major axis. Show that, where and are the two vertices of the ellipse.

Solution.

The equation of the ellipse is

Let the co-ordinates of P be

The co-ordinates of the foot of the perpendicular drawn from on the major axis at

and are the vertices of the ellipse,

and

Since the point lies on the ellipse

Hence, by and , we get

19. Show that for an ellipse the straight line joining the upper end of one latus rectum passes through the centre of the ellipse.

Solution.

Let the equation of the ellipse be

The co-ordinates of foci of the ellipse is

Now, the co-ordinates of the upper end of latus rectum passing through the focus is

Again, the co-ordinates of the lower end of latus rectum passing through the focus is

So, the mid-point of is

Hence, passes through the centre of the ellipse.

Find the equation of the auxiliary circle of the ellipse

21. If the eccentric angles of the two points on the ellipse are and then prove that the equation of the chord passing through these two points is

Solution.

By question, let and be two points on the given ellipse

Now, the equation of the chord of is

22. If the eccentric angles of the extremities of two chords which are passing through two points on major axis and the points are equidistant from centre of the ellipse, are respectively, then show that

Solution.

Let the equation of the ellipse be

Also, suppose that and are two chords of the ellipse whoch are equidistant from the centre of the ellipse. Consider that two chords intersect the major axis at and respectively.

Let the eccentric angles of the points be respectively.

Now, the slope of the slope of

Similarly, the slope of the slope of

Hence, from and we get,