Question

Two circles touch externally. The sum of their areas is $130\pi sq.$ $cm$ and the distance between their centres is $14cm$ . Find the radii of the circles.

Answer 1

If two circles touch externally, then the distance between their centres is equal to the sum of their radii.

Let the radii of the two circles be $r_{1}cm$ and $r_{2}cm$ respectively

Let $C_1$ and $C_2$ be the centres of the given circles. Then,

$C_1{C}_2=r_1+r_2$

$\Rightarrow 14=r_1+r_2$ $[\because C_1{C}_2=14cm$ given $]$

$\Rightarrow r_1+r_2=14$ ....(i)

It is given that the sum of the area of two circles is equal to $130\pi c{m}^2$.

$\therefore \pi r_1^2+\pi r_2^2=130\pi$

$\Rightarrow r_1^2+r_2^2=130$ ...(ii)

Now, $(r_1+r_2{)}^2=r_1^2+r_2^2+2r_1{r}_2$

$\Rightarrow {14}^2=130+2r_1{r}_2$ [ Using (i) and (ii)]

$\Rightarrow 196-130=2r_1{r}_2$

$\Rightarrow r_1{r}_2=33$ ...(iii)

Now, $(r_1-r_2{)}^2=r_1^2+r_2^2-2r_1{r}_2$

$\Rightarrow (r_1-r_2{)}^2=130-2\times 33$ [ using (ii) and (iii)]

$\Rightarrow (r_1-r_2{)}^2=64$

$\Rightarrow r_1-r_2=8$ ...(iv)

Solving, (i) and (iv), we get $r_1=11cm$ and $r_2=3cm$.

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