Question

Two circles touch internally. The sum of their areas is \[170\pi c{m^2}\] and the distance between their centres is \[4cm\]. Find the radii of the circles.

A. \[13cm,5cm\]
B. \[12cm,8cm\]
C. \[13cm,9cm\]
D. \[11cm,7cm\]

Answer 1

Hint: Consider two variables as the radii of the two circles. The distance between their radii is equal to the difference of the radii of the two circles. The area of the circle with radius \[rcm\] is given by \[\pi {r^2}c{m^2}\]. So, use this concept to reach the solution of the problem.

Complete step-by-step answer:

Let \[R\] be the radius of circle A and \[r\] be the radius of circle B.

Given distance between their centres is 4cm i.e., \[R - r = 4cm..................\left( 1 \right)\]
Area of the circle A = \[\pi {R^2}c{m^2}\]
Area of the circle B = \[\pi {r^2}c{m^2}\]

Since, the sum of the areas of the circles is \[170\pi c{m^2}\], we have \[\pi {R^2} + \pi {r^2} = 170\pi .................\left( 2 \right)\]

From (1) we have \[R = 4 + r\]

Substituting the value of \[R\] in equation (2), we get
\[
   \Rightarrow \pi {\left( {4 + r} \right)^2} + \pi {r^2} = 170\pi \\
   \Rightarrow \pi \left\{ {{{\left( {4 + r} \right)}^2} + {r^2}} \right\} = 170\pi \\
   \Rightarrow \left( {16 + {r^2} + 8r} \right) + {r^2} = 170 \\
   \Rightarrow 16 + {r^2} + 8r + {r^2} = 170 \\
   \Rightarrow 2{r^2} + 8r - 154 = 0 \\
\]

Splitting the terms, we have
\[
   \Rightarrow 2{r^2} + 22r - 14r - 154 = 0 \\
   \Rightarrow 2r\left( {r + 11} \right) - 14\left( {r + 11} \right) = 0 \\
   \Rightarrow \left( {2r - 14} \right)\left( {r + 11} \right) = 0 \\
   \Rightarrow r = 7, - 11 \\
  \therefore r = 7cm \\
\]

From equation (1), we have
\[
   \Rightarrow R - 7 = 4 \\
   \Rightarrow R = 7 + 4 = 11 \\
  \therefore R = 11cm \\
\]

Therefore, the radius of the circles is 11cm and 7cm.

Thus, the correct option is D, \[11cm,7cm\]

Note: Here we neglected the negative value of radius as radius cannot be negative. In the given question we can easily say that the radius of circle A is greater than the radius of circle B by the given diagram. Don’t forget to write the units after the areas of the circles and radii of the circle i.e., \[c{m^2}{\text{ & }}cm\] respectively.