Let the foci of a hyperbola be (-1, 2) and (9, 2) . If the length of its latus rectum is 9 2 , then the distance between its directrices is :

Step-by-step solution

The distance between the foci is 2ae . Given the foci are (-1, 2) and (9, 2) , the distance is (9 - (-1))^2 + (2 - 2)^2 = 10 . Thus, 2ae = 10 ae = 5 . The length of the latus rectum is given by 2b^2 a = 9 2 . For a hyperbola, we know that b^2 = a^2(e^2 - 1) = a^2e^2 - a^2 . Substituting ae = 5 , we get b^2 = 25 - a^2 . Now, substituting b^2 into the latus rectum equation: 2(25 - a^2) a = 9 2 4(25 - a^2) = 9a 100 - 4a^2 = 9a 4a^2 + 9a - 100 = 0 4a^2 + 25a - 16a - 100 = 0 a(4a + 25) - 4(4a + 25) = 0 (a - 4)(4a + 25)