If a tangent of slope 2 to the ellipse x^2 a^2 + y^2 b^2 =1 touches the circle x^2+y^2=4 , then maximum value of a b is

Step-by-step solution

Given the ellipse, x^2 a^2 + y^2 b^2 =1 Since, tangent to the given ellipse with slope m=2 is given as y=m x a^2 m^2+b^2 y=2 x 4 a^2+b^2 it touches the circle x^2+y^2=4 So, 2= 4 a^2+b^2 1+4 [distance from origin to the tangent] aligned & 4 a^2+b^2 =2 5 4 a^2+b^2=20 & A M . G M . 4 a^2+b^2 2 4 a^2 b^2 & 10 2 a b a b 5 aligned So, maximum value of a b=5 .