JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Circle

The first circle has centre $(1,1)$ and radius $1$; the second has centre $(9,1)$ and radius $2$. For the line $3x+4y-\lambda =0$ to keep the two circles on opposite sides, each centre must lie on the opposite side of the line and, further, the line must not cut into either circle. Opposite sides requires $(7-\lambda )(31-\lambda )<0$, i.e. $7<\lambda <31$. Staying clear of the first circle needs $\frac{|7-\lambda |}{5}\ge 1$, so $\lambda \ge 12$ within that range, and staying clear of the second needs $\frac{|31-\lambda |}{5}\ge 2$, so $\lambda \le 21$. Intersecting all of these gives $\lambda \in [12,21]$.