Question

Two circles which are not congruent touch externally. The sum of their areas is 130$\mathrm{\pi }$ cm² and the distance between the centres is 14 cm. Find the radii of the circles.

Answer 1

The circles touch externally at point P.
The distance between their centres is OO'.
Suppose that the radius of the first circle is r₁ and the radius of the second circle is r₂
Then OO' = 14 cm
$\Rightarrow$OP + O'P =14 cm
$\Rightarrow$r₁ + r₂ = 14 cm
$\Rightarrow r_2=14-r_1$ ....(i)
$$\begin{gathered} \mathrm{Now},\mathrm{sum}\mathrm{of}\mathrm{their}\mathrm{areas}=130\mathrm{\pi }{\mathrm{cm}}^2 \\ \Rightarrow \mathrm{\pi }{r_{\mathit{1}}}^{\mathit{2}}+\mathrm{\pi }{r_2}^2=130\mathrm{\pi } \\ \Rightarrow {r_{\mathit{1}}}^{\mathit{2}}+{r_2}^2=130 \end{gathered}$$
Putting the value of r_2, from(i) we get,
$$\begin{gathered} {r_1}^2+{\left(14-r_1\right)}^2=130 \\ \Rightarrow {r_1}^2+196+{r_1}^2-28r_1=130 \\ \Rightarrow 2{r_1}^2-28r_1+66=0 \\ \Rightarrow {r_1}^2-14r_1+33=0 \\ \Rightarrow {r_1}^2-11r_1-3r_1+33=0 \\ \Rightarrow r_1\left(r_1-11\right)-3\left(r_1-11\right)=0 \\ \Rightarrow \left(r_1-11\right)\left(r_1-3\right)=0 \\ \Rightarrow \left(r_1-11\right)=0\mathrm{or}\left(r_1-3\right)=0 \\ \Rightarrow r_1=3,11 \end{gathered}$$

Hence, the radii of the circles are 11 cm and 3 cm.