Let the ellipse x^2 a^2 + y^2 b^2 =1 contains the circle (x-1)^2+y^2=1 and has least area. If a^2+b^2=2 n , then find the value of n N .

Step-by-step solution

Since the ellipse contains the circle Solving circle with ellipse, we get aligned b^2 x^2+a^2 (1-(x-1)^2 ) & =a^2 b^2 (b^2-a^2 ) x^2+2 a^2 x-a^2 b^2 & =0 D & =0 4 a^4+4 (a^2 b^2 ) (b^2-a^2 ) & =0 a^2+b^2 (b^2-a^2 ) & =0 a^2-b^2 (a^2 e^2 ) & =0 1 & =b^2 e^2 b e=1 aligned Now, area of ellipse A= a b Now, aligned & A^2= ^2 a^2 b^2 & e^2=1- b^2 a^2 = 1 b^2 & e^2=1- 1 b^2 = b^2 a^2 & a^2= b^4 b^2-1 aligned Now, A^2=f(b)= b^6 b^2-1 For maxima and minima f^ (b)=0 aligned & (b^2-1 ) 6 b^5-b^6(2 b)=0 3 (b^2-1 )=b^2 & b^2= 3