##### In the previous article , we have solved few Short Answer Type questions of Hyperbola chapter of S.N.Dey mathematics, Class 11. In this article , we will solve few more Short Answer Type questions of Hyperbola related problems of S N Dey mathematics class 11.

6. **Find the equation of the hyperbola, referred to its axes as lines parallel to co-ordinate axes and having centre at eccentricity and the length of latus rectum .**

Solution.

The equation of the hyperbola having centre at and axes parallel to co-ordinate axes can be written as

The length of latus rectum

Hence, by the required equation of hyperbola is given by

7. **The distance between the vertices of a hyperbola is half the distance between its foci and the length of its semi-conjugate axis is . Referred to its axes as axes of co-ordinates, find the equation of the hyperbola. **

Solution.

We know that the distance between the vertices of the hyperbola is unit and the distance between the foci is unit.

By question,

Again, the length of semi-conjugate axis

Now,

Hence, the equation of the required hyperbola is

8. **The co-ordinates of the foci of a hyperbola are and and its eccentricity is find the equation of the hyperbola and the length of its latus rectum.**

Solution.

Now, the centre of the hyperbola is the mid-point of the foci i.e.,

From the co-ordinates of the foci, we can notice that the transverse axis of the hyperbola is along -axis and so the equation of the hyperbola having centre at is

9. **The centre of a hyperbola is at , the co-ordinates of one of its vertices are and eccentricity is find its equation.**

Solution.

If is the distance of the vertex from the centre , then

Again,

Since the ordinates of the centre and the vertex of the hyperbola is , both cantre and the vertex lie on the straight line

So, the transverse axis of the hyperbola is parallel to -axis. So, the equation of the hyperbola with centre is

10. *Find the equation of the hyperbola with centre at the origin, transverse axis on -axis, passing through the point and having the length of semi-conjugate axis *

Solution.

The hyperbola having centre at the origin and transverse axis on -axis, can be written in the form of

Since the hyperbola passes through the point

Hence, putting the values of and in , we get the required equation of hyperbola which is given by

11. *The lengths of transverse and conjugate axes of a hyperbola are unit and unit respectively and their equations are and find the equation of the hyperbola.*

Solution.

It is given that

Since the transverse and conjugate axes of the hyperbola are and , the point of intersection of and is given by which is the centre of the hyperbola.

Since the transverse axis of the hyperbola given by is parallel to -axis, so the equation of the hyperbola with centre is given by

12. * and are respectively the eccentricities of a hyperbola and its conjugate. Prove that, *

Solution.

Let the equation of the hyperbola be

The equation of the conjugate hyperbola of can be written as

13. **Find the eccentricity of the hyperbola If are the foci and any point on this hyperbola, prove that, ( is the origin).**

Solution.

Comparing with the general equation of hyperbola we get

The co-ordinates of foci of the hyperbola are i.e.,

The co-ordinates of any point on the hyperbola can be taken as

Let so that

Now,

Hence by and , the result follows.

14. * is a variable point on the hyperbola and is a fixed point. Show that the locus of the middle point of the line segment is another rectangular hyperbola.*

Solution.

The co-ordinates of any point on the hyperbola can be written as

Let the middle point of be

From and we get,

So, by we can say that the locus of the middle point of the line segment is

Since the equation is a rectangular hyperbola, so the result follows.

15. *Show that the difference of focal distances of any point of the hyperbola is constant.*

Solution.

Comparing with the standard form of the hyperbola , we get

Now, the co-ordinates of foci are given by

Let any point on the hyperbola be

Similarly, we can easily show that

Hence, the difference of focal distances of any point of the given hyperbola is constant.

16. *The co-ordinates of the foci of a hyperbola are and the length of its latus rectum is unit; find its equation.*

Solution.

By question, we notice that the co-ordinates of the foci lie on the -axis. Now, the transverse axis of the hyperbola is along -axis.

The equation of the hyperbola can be written as

Now,

So, the equation of the hyperbola is given by

The eccentricity of the hyperbola is

17. *Show that for all values of the point lies on a fixed hyperbola. What is the value of the eccentricity of the hyperbola.*

Solution.

Clearly, the equation represents a hyperbola whose eccentricity is given by