Question

Two circles touch internally. The sum of their areas is $116\pi c{m}^2$ and distance between their centres is $6cm$. Find the radii of the circles.

• A. $5cm$ & $8cm$
• B. $10cm$ & $5cm$
• C. $18cm$ & $4cm$
• D. None of these

Answer 1

The correct option is D None of these

Let R and r be the radii of the circles having centres at O and O' respectively. Then,
Sum of areas $=116\pi c{m}^2$
$\Rightarrow \pi R^2+\pi r^2=116\pi$
$R^2+r^2=116$ ....(i)
Distance between the centres $=6cm$
$\Rightarrow O{O}^{\prime }=6cm$
$\Rightarrow R-r=6$ ....(ii)
Now,
$(R+r{)}^2+(R-r{)}^2=2(R^2+r^2)$
$\Rightarrow (R+r{)}^2+36=2\times 116$ [using (i) and (ii)]
$\Rightarrow (R+r{)}^2=(2\times 116-36)=196$
$\Rightarrow R+r=14$ ...(iii)
Solving (ii) and (iii), we get $R=10$ and $r=4$.
Hence, radii of the given circles are $10cm$ and $4cm$ respectively.