Let H: x^2 a^2 - y^2 b^2 =1 be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is 8 3 . If the line x= intersects the hyperbola H at the points A and B such that the area of the triangle AOB is 4 15 , where O is the origin, then ^2 equals

Step-by-step solution

Given distance between foci is 2ae = 6 ae = 3 Distance between directrices is 2a e = 8 3 Multiplying the two equations, we get 4a^2 = 16 a^2 = 4 Dividing the two equations, we get e^2 = 9 4 Using b^2 = a^2(e^2 - 1) , we get b^2 = 4 ( 9 4 - 1 ) = 5 The equation of the hyperbola is x^2 4 - y^2 5 = 1 The line x = intersects the hyperbola at A and B . Substituting x = , we get ^2 4 - y^2 5 = 1 y = 5( ^2 - 4) 2 The coordinates of A and B are ( , 5( ^2 - 4) 2 ) and ( , - 5( ^2 - 4) 2 ) The length of the base AB is 5( ^2