If the distance between the foci and the distance between the two directrixes are in the ratio 3: 2 for a hyperbola x^2 a^2 - y^2 b^2 =1 , then a : b is

Step-by-step solution

For the hyperbola x^2 a^2 - y^2 b^2 = 1 , the distance between the foci is 2ae and the distance between the directrices is 2a e . Given the ratio of these distances is 3:2 , we have: 2ae 2a/e = 3 2 e^2 = 3 2 Using the relation b^2 = a^2(e^2 - 1) , we substitute e^2 = 3 2 : b^2 = a^2 ( 3 2 - 1 ) = a^2 ( 1 2 ) b^2 a^2 = 1 2 b a = 1 2 Therefore, a b = 2 : 1 . Answer: 2 : 1