Question 1

What is the distance between the foci of the hyperbola x^2 - 4y^2 = 1?

Accepted Answer

The given equation of the hyperbola is x^2 - 4y^2 = 1. This can be rewritten in the standard form as x^2 1 - y^2 1/4 = 1. Comparing with x^2 a^2 - y^2 b^2 = 1, we get a^2 = 1 and b^2 = 1 4 . The eccentricity e of the hyperbola is given by e = 1 + b^2 a^2 . e = 1 + 1/4 1 = 5 4 = 5 2 The distance between the foci of the hyperbola is 2ae. 2ae = 2 1 5 2 = 5 Answer: 5

Question 2

What is the distance between the two foci of the hyperbola 25x^2 - 75y^2 = 225?

Accepted Answer

The given equation of the hyperbola is 25x^2 - 75y^2 = 225. Dividing the entire equation by 225, we get: x^2 9 - y^2 3 = 1 Comparing this with the standard equation of a hyperbola x^2 a^2 - y^2 b^2 = 1, we have a^2 = 9 and b^2 = 3. The relation between a, b, and the distance from the center to a focus c is given by c^2 = a^2 + b^2. Substituting the values, we get: c^2 = 9 + 3 = 12 c = 12 = 2 3 The distance between the two foci is 2c. Distance = 2(2 3 ) = 4 3 units. Answer: 4 3 units

Hyperbola — Previous Year Questions & Solutions