Two circles touch internally. The sum of their areas is `116 pi cm^(2)` and distance between their centres is 6 cm. Find the radii of the circles

Answer

To solve the problem of finding the radii of two circles that touch internally, we can follow these steps:

### Step 1: Define Variables
Let:
- \( R \) = radius of the larger circle
- \( r \) = radius of the smaller circle

### Step 2: Write the Equation for the Sum of Areas
The area of a circle is given by the formula \( \text{Area} = \pi r^2 \). Therefore, the sum of the areas of the two circles can be expressed as:
\[ \pi R^2 + \pi r^2 = 116\pi \]
Dividing through by \( \pi \):
\[ R^2 + r^2 = 116 \quad \text{(Equation 1)} \]

### Step 3: Write the Equation for the Distance Between Centers
Since the circles touch internally, the distance between their centers is given by:
\[ R - r = 6 \quad \text{(Equation 2)} \]

### Step 4: Solve Equation 2 for \( R \)
From Equation 2, we can express \( R \) in terms of \( r \):
\[ R = r + 6 \]

### Step 5: Substitute \( R \) in Equation 1
Substituting \( R = r + 6 \) into Equation 1:
\[ (r + 6)^2 + r^2 = 116 \]
Expanding \( (r + 6)^2 \):
\[ r^2 + 12r + 36 + r^2 = 116 \]
Combining like terms:
\[ 2r^2 + 12r + 36 = 116 \]
Subtracting 116 from both sides:
\[ 2r^2 + 12r - 80 = 0 \]

### Step 6: Simplify the Quadratic Equation
Dividing the entire equation by 2:
\[ r^2 + 6r - 40 = 0 \]

### Step 7: Factor the Quadratic Equation
To factor \( r^2 + 6r - 40 = 0 \), we look for two numbers that multiply to \(-40\) and add to \(6\). The numbers \(10\) and \(-4\) work:
\[ (r + 10)(r - 4) = 0 \]
Setting each factor to zero gives:
\[ r + 10 = 0 \quad \Rightarrow \quad r = -10 \quad \text{(not valid)} \]
\[ r - 4 = 0 \quad \Rightarrow \quad r = 4 \]

### Step 8: Find \( R \)
Now that we have \( r = 4 \), we can find \( R \):
\[ R = r + 6 = 4 + 6 = 10 \]

### Final Answer
The radii of the circles are:
- Radius of the larger circle \( R = 10 \, \text{cm} \)
- Radius of the smaller circle \( r = 4 \, \text{cm} \)