Question

If A is the minimum area of the circle which touches the parabolas $y=x^2+1$ and $y^2=x-1$, then the value of $\frac{32}{\pi }A$ is ___

Answer 1

$y=x^2+1,x=y^2+1$ are symmetrical abut y = x.
$\Rightarrow$ slope of tangent $=1=\frac{dy}{dx}$
$\Rightarrow A(\frac{1}{2},\frac{5}{4}),B(\frac{5}{4},\frac{1}{2})$ are the end points of a diameter. $\Rightarrow r=\frac{3}{8}\sqrt{2}$
$\Rightarrow Area=\frac{9\pi }{32}$