###### In the previous articles , we have done complete solution of Ellipse chapter of S.N.Dey mathematics, Class 11. In this article , we will solve few questions of Hyperbola related problems.

1.Explain what general diagrams are represented by the equation where and are constants.

Solution.

We have the equation given by

Now, there can be three conditions for different values of

Case

For

In this case, we get from

Clearly, represents the equation of the ellipse.

Case

For we get by

Equation represents a ‘circle.’

Case

For

Let

So, from we get,

Equation represents a ‘hyperbola.’

2. Show that the eccentricity of any rectangular hyperbola is

Solution.

We know that the general form of hyperbola is

The eccentricity of is given by

For rectangular hyperbola, and so from we get,

3. Find the eccentricity, co-ordinates of the foci and the equations of the directrices of the hyperbolas :

Solution.(i)

Comparing with the general form of hyperbola , we get

Co-ordinates of foci :

The equations of directrices of the hyperbola :

Solution(ii)

Comparing with the general form of hyperbola , we get

Co-ordinates of foci :

The equations of directrices of the hyperbola :

4. Show that the eccentricities of the two hyperbolas and are equal.

Solution.

We have two hyperbolas given by

and .

Comparing with the general form of hyperbola , we get

Comparing with the general form of hyperbola , we get

Note[*] : eccentricity of the first hyperbola, eccentricity of the second hyperbola.

Find the co-ordinates of the foci of the hyperbola

Solution.

Comparing with the general form of hyperbola , we get

The co-ordinates of foci of the hyperbola :

Find the eccentricity and the length of latus rectum of the hyperbola

Solution.

Comparing with the general form of hyperbola , we get

So,

The length of the latus rectum is

6. Find the length of latus rectum and the equations of the directrices of the hyperbola

Solution.

Comparing with the general form of hyperbola , we get

The length of the latus rectum of the hyperbola is

Equations of the directrices of the hyperbola

7. What type of conic is represented by the equation What is its eccentricity?

Solution.

The given equation represents the equation of rectangular hyperbola and its eccentricity is

8. If the length of conjugate axis and the length of latus rectum of a hyperbola are equal, find its eccentricity.

Solution.

The length of the conjugate axis unit and the length of the latus rectum is unit of the hyperbola

By question,