Question

A circle of radius $2$ has center at $(2,0)$. Another circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. Which of the following is the y-intercept of the line?

• A. $3$
• B. $\frac{\sqrt{2}}{4}$
• C. $\frac{8}{3}$
• D. $2\sqrt{2}$

Answer 1

The correct option is D $2\sqrt{2}$

$\begin{array}{c}\frac{C_{1}P}{C_{2}P}=\frac{2}{1}\\ \left(\frac{2\times 5-2\times 1}{2-1},\frac{2\times 0-1\times 0}{2-1}\right)\\ P=\left(8,0\right)\end{array}$

Equation of line through $(8,0)$

$y-0=m(x-8)$

$y-mx+8m=0$

$\begin{array}{c}\frac{\left(-5m+8m\right)}{\sqrt{1+m^2}}=1\\ 9m^2=1+m^2\\ m^2=\frac{1}{8}\\ m=\pm \frac{1}{2\sqrt{2}}\end{array}$