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
B. 16
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 - 4) and the height of the triangle AOB from the origin is | | Area of AOB = 1 2 AB | | = 4 15 1 2 5( ^2 - 4) | | = 4 15 Squaring both sides, we get 1 4 5( ^2 - 4) ^2 = 240 ^4 - 4 ^2 - 192 = 0 ( ^2 - 16)( ^2 + 12) = 0 Since ^2 > 0, we have ^2 = 16 Answer: 16
Related: Mathematics — Hyperbola · All PYQ Banks