Question

The set of all real values of X 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. (12, 32)
• B. (18, 42)
• C. (12, 24)
• D. (18, 48)

Answer 1

Answer: (B)

$C_{1} \left(2, 2\right)C_{2} \left(5, 5\right)$
$r_{1} = \sqrt{2} r_{2} = \sqrt{50}-1$
$\left|r_{1}-r_{2}\right| < c_{1}\,c_{2} < r_{1}+r_{2}$
$\left|\sqrt{50-\lambda}-\sqrt{2}\right|< \sqrt{9+9} < \sqrt{50-\lambda }+\sqrt{2}$
$-18 <\left[\sqrt{50-\lambda }-\sqrt{2}\right] < 18\quad\quad\sqrt{18}-\sqrt{2} < \sqrt{50-\lambda }$
$\quad\quad \lambda > 18\quad\quad20-12 < 50-\lambda$
$\quad\quad\quad \quad \quad \quad \quad \quad \lambda < 42$
$\lambda \in \left(18,\,42\right)$