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 the radius of the larger circle be \( R_1 \) and the radius of the smaller circle be \( r \).

### Step 2: Set Up the Equations
We know two things from the problem:
1. The sum of the areas of the circles is \( 116\pi \) cm².
2. The distance between the centers of the circles is \( 6 \) cm.

The area of a circle is given by the formula \( \pi r^2 \). Therefore, we can write the first equation as:
\[ \pi R_1^2 + \pi r^2 = 116\pi \]
Dividing through by \( \pi \) gives:
\[ R_1^2 + r^2 = 116 \quad \text{(1)} \]

For the second condition, since the circles touch internally, the distance between their centers is equal to the difference of their radii:
\[ R_1 - r = 6 \quad \text{(2)} \]

### Step 3: Solve for One Variable
From equation (2), we can express \( R_1 \) in terms of \( r \):
\[ R_1 = r + 6 \quad \text{(3)} \]

### Step 4: Substitute in the First Equation
Now, substitute equation (3) into equation (1):
\[ (r + 6)^2 + r^2 = 116 \]
Expanding the left side:
\[ (r^2 + 12r + 36) + r^2 = 116 \]
Combining like terms:
\[ 2r^2 + 12r + 36 = 116 \]
Subtracting \( 116 \) from both sides:
\[ 2r^2 + 12r - 80 = 0 \]

### Step 5: Simplify the Quadratic Equation
Dividing the entire equation by \( 2 \):
\[ r^2 + 6r - 40 = 0 \]

### Step 6: Factor the Quadratic
Now we can factor the quadratic:
\[ (r + 10)(r - 4) = 0 \]
Setting each factor to zero gives us:
\[ r + 10 = 0 \quad \Rightarrow \quad r = -10 \quad \text{(not valid)} \]
\[ r - 4 = 0 \quad \Rightarrow \quad r = 4 \]

### Step 7: Find the Radius of the Larger Circle
Using \( r = 4 \) in equation (3):
\[ R_1 = 4 + 6 = 10 \]

### Step 8: Conclusion
Thus, the radii of the circles are:
- Radius of the larger circle, \( R_1 = 10 \) cm
- Radius of the smaller circle, \( r = 4 \) cm

### Final Answer
The radii of the circles are \( 10 \) cm and \( 4 \) cm.

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