Minimum value of the expression $(a-b)^2 + \left(4\sqrt{1+a^2} + 2\sqrt{2b-b^2}\right)^2$ is
Step-by-Step Solution
Key Concept: Interpret as the square of the distance between a point on the curve $y=-4\sqrt{1+x^2}$ (lower branch of hyperbola $y^2/16-x^2=1$) and a point on the semicircle $y=2\sqrt{1-(x-1)^2}$ (upper half of circle centre $(1,0)$, radius $2$). Find minimum distance.
Minimum distance between the two curves is $2$, so minimum of expression $= 4$.
Correct Answer: (B) 4