The set of all real values of for which exactly two common tangents can be drawn to the ci…

Question

The set of all real values of $\lambda$ for which exactly two common tangents can be drawn to the circles $x^2+y^2-4x-4y+6=0$ and $x^2+y^2-10x-10y+\lambda=0$ is the interval:

Accepted Answer

Circle 1: centre $(2,2)$, $r_2=\sqrt2$. Circle 2: centre $(5,5)$, $r_1=\sqrt{50-\lambda}$.

$d=\sqrt{9+9}=3\sqrt2$. Exactly two common tangents requires $|r_1-r_2|<d<r_1+r_2$:
$$\sqrt{50-\lambda}-\sqrt2<3\sqrt2<\sqrt{50-\lambda}+\sqrt2.$$
This simplifies to $2\sqrt2<\sqrt{50-\lambda}<4\sqrt2\Rightarrow 8<50-\lambda<32\Rightarrow 18<\lambda<42.$

Solutions · 1