JEE Main 2017 · 2 Apr · Shift 1, Mathematics question 79

The intended answer takes the circle on the $y$-axis, centre $(0,k)$, touching $y=|x|$ (so $r=\frac k{\sqrt2}$) and touching the parabola at its vertex ($r=4-k$). That gives $r=4(\sqrt2-1)$.

But at its vertex the parabola has radius of curvature $\frac12$, smaller than this $r$, so such a circle would cross the parabola near the vertex instead of touching it. The question was given marks to all.