Question

Two circles touch each other externally. The sum of their area is $58$ $\pi c{m}^2$ and the distance between their center is $10cm$. Find the radii of the two circles.

• A. $7cm$and $3cm$
• B. $5cm$and $7cm$
  
• C. $3cm$and $2cm$
  
• D. $Noneofthese$

Answer 1

The correct option is A $7cm$and $3cm$

Let r be the radius of one circle and,
$10-r$ be the radius of other circle.
The area of the first circle is $\pi r^2$
The area of the first circle is $\pi (10-r^2)$
Given,
The sum of these areas is $58\pi c{m}^2$
Therefore,
$\pi r^2+\pi (10-r{)}^2=58\pi$
$=>r^2+(10-r{)}^2=58$ [dividing both sides by $\pi$]
$=>r^2+(10{)}^2+(r{)}^2-2\left(10\right)\left(r\right)=58$
$=>r^2+100+r^2-20r=58$
$=>2r^2-20r+42=0$
$=>r^2-10r+21=0$ [dividing both sides by $2$]
$=>r^2-7r-3r+21=0$
$=>r(r-7)-3(r-7)=0$
$=>(r-7)(r-3)=0$
$r=3cm$ and $r=7cm$