Let H ₁: x^2 a ^2 - y^2 ~b ^2 =1 and H ₂:- x^2 ~A ^2 + y^2 ~B ^2 =1 be two hyperbolas having length of latus rectums 15 2 and 12 5 respectively. Let their ecentricities be e₁= 5 2 and e₂ respectively. If the product of the lengths of their transverse axes is 100 10 , then 25 e ₂^2 is equal to ________.

Step-by-step solution

x^2 a^2 - y^2 b^2 =1 2 b^2 a =15 2 ...(i) 1+ b^2 a^2 = 5 2 ....(ii) From (i) and (ii) a=5 2 and b^2=75 x^2 A^2 - y^2 B^2 =-1 2 A^2 B =12 5 ....(iii) Since, product of transverse axis is =100 10 (2 A) (2 B)=100 10 From (iii) and (iv) aligned & A^2=150 and B=5 5 & e₂= 1+ A^2 B^2 = 11 5 & 25 e₂^2=25 ( 11 5 )=55 aligned