Let the product of the focal distances of the point P (4,2 3 ) on the hyperbola H : x ^2 a ^2 - y ^2 ~b ^2 =1 be 32 . Let the length of the conjugate axis of H be p and the length of its latus rectum be q . Then p ^2+ q ^2 is equal to _______

Step-by-step solution

x^2 a^2 - y^2 b^2 =1 ....(1) aligned & P (4,2 3 ) & PS ₁ PS ₂=32 & | PS ₁- PS ₂ |=2 a & P (4,2 3 ) lies on H & 16 a ^2 - 12 ~b ^2 =1 aligned 16 b^2-12 a^2=a^2 b^2 ....(2) aligned & | PS ₁- PS ₂ |^2=4 a ^2 & PS ₁^2+ PS ₂ ^2-2 PS ₁ PS ₂=4 a ^2 & ( ae -4)^2+12+( ae +4)^2+12-64=4 a ^2 & 2 a ^2 e ^2-8=4 a ^2 & a ^2+ b ^2-4=2 a ^2 & ~b ^2- a ^2=4 aligned aligned & (2) &(3) 16 (a^2+4 )-12 a^2=a^2 (a^2+4 ) & 16 a^2+64-12 a^2=a^4+4 a^2 & a^4=64 & a^2=8 aligned aligned & b ^2=12 & p ^2+ q ^2=4 ~b ^2+ 4 ~b ^4 a ^2 & =120 alig