Let e₁ and e₂ be the eccentricities of the ellipse x ^2 ~b ^2 + y ^2 25 =1 and the hyperbola x ^2 16 - y ^2 ~b ^2 =1 , respectively. If b 5 and e ₁ e ₂=1 , then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

Step-by-step solution

aligned & e ₁^2=1- b ^2 25 e ₂^2=1- b ^2 16 & e ₁^2 e ₂^2=1 & (1- b ^2 25 ) (1+ b ^2 16 )=1 & 2+ b ^2 16 - b ^2 25 - b ^2 400 =1 & 9 ~b ^2 400 = b ^4 400 & ~b ^2=9 & x ^2 9 + y ^2 25 =1 x ^2 16 - y ^2 9 =0 & e ₁ 1- 9 25 & e ₁= 4 5 aligned Focii : - (0, 4) ( 5,0) ellipse passing through all four foci aligned & x^2 25 + y^2 16 =1 & e= 1- 16 25 = 3 5 aligned