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 :

• A. $(18,42)$
• B. $(12,32)$
• C. $(12,24)$
• D. $(18,48)$

Answer 1

The correct option is A $(18,42)$

Two circles can have exactly two common tangents only if the circles intersect each other. For the circles to intersect each other, the following conditions must hold true -
$r_1+r_2>dist(c_1,c_2)$
$r_2-r_1<dist(c_1,c_2)$
$\sqrt{2}+\sqrt{50-\lambda }>3\sqrt{2}\Rightarrow 50-\lambda <8\Rightarrow \lambda <42$
$\sqrt{50-\lambda }-\sqrt{2}<3\sqrt{2}\Rightarrow 50-\lambda <32\Rightarrow \lambda >18$

$\therefore \lambda$ lies in the interval $(18,24)$