JEE Main 2014 · 11 Apr · Shift 1, Mathematics question 80

The circles are $C_1(2,2)$, $r_1=\sqrt2$ and $C_2(5,5)$, $r_2=\sqrt{50-\lambda}$, with $d=3\sqrt2$.

Exactly two common tangents means they intersect: $|r_2-r_1|<d<r_1+r_2$. This gives $2\sqrt2<r_2<4\sqrt2$, so $8<50-\lambda<32$ and $\lambda\in(18,42)$.