Question

The sum of the areas of two circles, which touch each other externally, is $153\pi$. If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

• A. 4
• B. 2
• C. 3
• D. None of these

Hint:

Use sum and product of radii to find ratio via quadratic in $k = R/r$.

Answer 1

The Correct Option is C

Let radii = $R$ and $r$. $R + r = 15$ and $\pi(R^2 + r^2) = 153\pi \Rightarrow R^2 + r^2 = 153$. $(R + r)^2 = R^2 + r^2 + 2Rr \Rightarrow 225 = 153 + 2Rr \Rightarrow 2Rr = 72 \Rightarrow Rr = 36$. $\frac{R}{r} + \frac{r}{R} = \frac{R^2 + r^2}{Rr} = \frac{153}{36} = 4.25$. Also $\frac{R}{r} + \frac{r}{R} = k + \frac{1}{k} = 4.25 \Rightarrow k^2 - 4.25k + 1 = 0$. Solving gives $k = 3$ or $1/3$. Ratio = 3:1.