Question

Two circle touch internally. The sum of there areas is $170$ $\pi c{m}^2$ and the distance between their centres is $4$ cm. Find the radii of the circles

• A. $13$ cm, $5$ cm
• B. $12$ cm, $8$ cm
• C. $13$ cm, $9$ cm
• D. $11$ cm, $7$ cm

Answer 1

The correct option is D $11$ cm, $7$ cm

Let radius of circle $A$ be $r_1$ and radius of circle $B$ be $r_2$
Given, $r_1-r_2=4$ ---------(i)
and $\pi {r_1}^2+\pi {r_2}^2=170\pi$
$\Rightarrow {r_1}^2+{r_2}^2=170$ -----------(ii)
(i) $r_1=4+r_2$
Now, putting this value in (ii), we get
$\Rightarrow (4+r_2{)}^2+{r_2}^2=170$
$\Rightarrow 16+8r_2+{r_2}^2+{r_2}^2=170$
$\Rightarrow 2{r_2}^2+8r_2-154=0$
$\Rightarrow 2{r_2}^2+22r_2-14r_2-154=0$
$\Rightarrow 2r_2(r_2+11)-14(r_2+11)=0$
$\Rightarrow (r_2+11)\left(\right(2r_2-14\left)\right)=0$
$\Rightarrow r_2=-11$ (not possible) or $r_2=7$
Thus, $r_2=7$
$r_1=r_2+4$
$\Rightarrow r_1=7+4=11$
Thus, the radius of the two circles are $11$ cm and $7$ cm.