Question

Two circles touch externally. The sum of their areas is $117$ $\pi c{m}^2$ and the distance between their centres is $15$ cm. Find the radii of the two circles.

• A. $9$ cm, $6$ cm
• B. $10$, $7$ cm
• C. $12$ cm, $6$ cm
• D. $8$ cm, $8$ cm

Answer 1

The correct option is A $9$ cm, $6$ cm

Let radius of circle $A$ be $r_1$ and radius of circle $B$ be $r_2$
Given, $r_1+r_2=15$ -------(i)
and $\pi {r_1}^2+\pi {r_2}^2=117\pi$
$\Rightarrow {r_1}^2+{r_2}^2=117$ ------(ii)
(i)$\Rightarrow r_1+r_2=15$
$\Rightarrow r_1=15-r_2$
Now, putting this value in (ii), we get
$\Rightarrow (15-r_2{)}^2+{r_2}^2=117$
$\Rightarrow {15}^2-2\times 15\times r_2+{r_2}^2+{r_2}^2=117$
$\Rightarrow 225-30r_2+2{r_2}^2=117$
$\Rightarrow 2{r_2}^2-30r_2+108=0$
$\Rightarrow 2{r_2}^2-18r_2-12r_2+108=0$
$\Rightarrow 2r_2(r_2-9)-12(r_2-9)=0$
$\Rightarrow (r_2-9)(2r_2-12)=0$
$\Rightarrow r_2=9$ or $r_2=\frac{12}{2}=6$
$\Rightarrow r_1=15-9$ or $r_1=15-6$
$\Rightarrow r_1=6$ or $r_1=9$
$\Rightarrow r_1>r_2$
Thus, the radius of the two circles are $9$ cm and $6$ cm.