Question

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

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

Answer 1

The correct option is D 24 cm

The distance between centres of 2 circles = 6 cm.
Sum of areas of 2 circles = 180 $\pi$
$\pi r_1^2+\pi r_2^2=180\pi \Rightarrow r_1^2+r_2^2=180\dots \left(i\right)$

and distance between centres i.e.
$r_1-r_2=6\Rightarrow r_2=r_1-6\dots \left(ii\right)$

From eq. (i) we get,
$r_1^2+(r_1-6{)}^2=180$
$\Rightarrow r_1^2+(r_1^2-12r_1+36)=180\Rightarrow 2r_1^2-12r_1+36=180\Rightarrow 2r_1^2-12r_1-144=0\Rightarrow r_1^2-6r_1-72=0\Rightarrow (r_1-12)(r_1+6)=0$

Hence, $r_1$ = 12 m and $d_1$ = 24 cm