JEE Main 2025 Apr 3 Shift 1, Mathematics Q18: Involving Two or More Circles

The two parabolas are reflections of one another in the line $y=x$. The smallest circle that touches both must therefore be centred on $y=x$ and touch each parabola at the point closest to the line of symmetry. At those touch points the tangents are parallel to $y=x$, i.e. have slope $1.$

On $y=x^2+2$: $\frac{dy}{dx}=2x=1\Rightarrow x=\frac{1}{2}$, giving the contact point $B=\left(\frac{1}{2},\frac{9}{4}\right).$

By symmetry the other contact point is $A=\left(\frac{9}{4},\frac{1}{2}\right).$

The chord $AB$ is a diameter of the smallest circle that simultaneously touches both:

$AB=\sqrt{{\left(\frac{9}{4}-\frac{1}{2}\right)}^2+{\left(\frac{1}{2}-\frac{9}{4}\right)}^2}=\sqrt{2}\cdot \left|\frac{7}{4}\right|=\frac{7\sqrt{2}}{4}.$

Radius $=\frac{AB}{2}=\frac{7\sqrt{2}}{8}.$