Mathematics - Parabola Question with Solution | TestHub

The given parabolas are symmetric about the line $y=x$

Tangents at $\mathrm{A}\&\mathrm{B}$ must be parallel to $\mathrm{y}=\mathrm{x}$ line, so slope of the tangents $=1$
${\left(\frac{dy}{dx}\right)}_{\min A}=1={\left(\frac{dy}{dx}\right)}_{\min B}$
For point $\mathrm{B},\mathrm{y}={\mathrm{x}}^2+2$
$\begin{array}{c}\frac{dy}{dx}=2x=1\\ x=\frac{1}{2}\Rightarrow y=\frac{9}{4}\\ \therefore \text{ Point }B=\left(\frac{1}{2},\frac{9}{4}\right)\Rightarrow \text{ Point }\mathrm{A}=\left(\frac{9}{4},\frac{1}{2}\right)\\ \mathrm{AB}=\sqrt{{\left(\frac{1}{2}-\frac{9}{4}\right)}^2+{\left(\frac{9}{4}-\frac{1}{2}\right)}^2}\\ =\sqrt{\frac{98}{16}}=\frac{7\sqrt{2}}{4}\end{array}$
Radius $==\frac{7\sqrt{2}}{8}$