# Hyperbola (S.N.Dey) |Part-1 | Ex-6

###### 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, 