Question

Two circles touch each other internally and their centres are $O$ and $O^{\prime }$ as shown. The sum of their areas is $180\pi$ sq. cm. and the distance between their centres is $6cm.$ What is the diameter of the larger circle?

• A. $16cm$
• B. $12cm$
• C. $18cm$
• D. $24cm$

Answer 1

The correct option is D $24cm$

The distance between centres of circles $=6cm$

Sum of areas of the circles $=180\pi$

Let the radius of the larger circle be $r_1$ and the radius of the smaller circle be $r_2$

$\Rightarrow \pi r_1^2+\pi r_2^2=180\pi$

$\Rightarrow r_1^2+r_2^2=180\dots \dots \left(i\right)$

and distance between their centres, i.e.

$r_1-r_2=6$

$\Rightarrow r_2=r_1-6\dots \dots \left(ii\right)$

From equations (i) and (ii), we get

$r_1^2+(r_1-6{)}^2=180$

$\Rightarrow r_1^2+(r_1^2-12r_1+36)=180$

$\Rightarrow 2r_1^2-12r_1+36=180$

$\Rightarrow 2r_1^2-12r_1-144=0$

$\Rightarrow r_1^2-6r_1-72=0$

$\Rightarrow r_1^2-12r_1+6r_1-72=0$

$\Rightarrow r_1(r_1-12)+6(r_1-12)=0$

$\Rightarrow (r_1-12)(r_1+6)=0$

$\Rightarrow r_1=12,r_1=-6$ [Taking positive value]

Hence, $r_1=12cm\Rightarrow d_1=12\times 2=24cm$