Question

The minimum area of circle which touches the parabolas $y=\frac{1}{2}\left(x^{2}+5\right)$ and $x=\frac{1}{2}\left(y^{2}+5\right)$ is

• A. $\pi$
• B. $2 \pi$
• C. $2 \sqrt{2} \pi$
• D. $\sqrt{2} \pi$

Answer 1

Answer: (B)

Parabolas, $y =\frac{1}{2}\left( x ^{2}+5\right)$ and $x =\frac{1}{2}\left( y ^{2}+5\right)$ are symmetrical about the line $y = x$
$\therefore$ tangent at $A$ is parallel to $y = x$
$y=\frac{1}{2}\left(x^{2}+5\right) $
$\frac{d y}{d x}=\frac{1}{2}(2 x)=1 \Rightarrow x=1$
co-ordinate of $A$ is $(1,3)$
$\therefore$ co-ordinate of $B$ is $(3,1)$
Radius of circle $( r )=\frac{1}{2}( AB )=\frac{1}{2} \sqrt{4+4}=\sqrt{2}$
Area of circle $=\pi r^{2}=2 \pi $