On the coordinate plane, ellipse C₁: x^2 a₁^2 + y^2 b₁^2 =1 (a₁>b₁>0 ) and hyperbola C₂: x^2 a₂^2 - y^2 b₂^2 =1 (a₂, b₂>0 ) has the same focus points F₁, F₂ . Point P is one of the intersection points of C₁ and C₂ and P F₁ P F₂ . If e₁ is the eccentricity of C₁ and e₂ is the eccentricity of C₂ . Then find the minimum value of 9 e₁^2+e₂^2 .

Step-by-step solution

Given aligned a₁ e₁ & =a₂ e₂ P F₁^2+P F₂^2 & =4 a₁^2 e₁^2...(1) Using Pythagorus theorem P F₁+P F₂ & =2 a₁ ...(2) [P F₁+P F₂ ] & =2 a₂ ...(3) (2)^2+(3)^2 & 2 [P F₁^2+P F₂^2 ]=24 (a₁^2+a₂^2 )...(4) aligned From Eqns. (1) and (4) aligned 2 (a₁^2+a₂^2 ) & =4 a₁^2 e₁^2 a₁^2+a₂^2 & =2 (a₁^2 e₁^2 ) 1+ ( a₂ a₁ )^2 & =2 e₁^2 1+ ( e₁ e₂ )^2 & =2 e₁^2 aligned aligned & 1 e₁^2 + 1 e₂^2 =2 & Let 1 e₁ = 2 , 1 e₂ = 2 & E=9 e₁^2+e₂^2= 9 2 ^2 + 1 2 cosec ^2 & = 1 2 [9 (1+ ^2 )+1+ ^2 ] & = 1 2 [10+9 ^2 + ^2 ] minimum value =6 using