Two circles touch each other externally. The sum of their areas is `74pi cm^(2)` and the distance between their centres is 12 cm. Find the diameters of the circle.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the radius of the first circle be \( r_1 \) and the radius of the second circle be \( r_2 \).

### Step 2: Set up the equations
According to the problem, the distance between the centers of the two circles is 12 cm. Therefore, we can write:
\[ r_1 + r_2 = 12 \quad \text{(Equation 1)} \]

The sum of the areas of the two circles is given as \( 74\pi \, \text{cm}^2 \). The area of a circle is given by the formula \( \pi r^2 \). Thus, we can write:
\[ \pi r_1^2 + \pi r_2^2 = 74\pi \]
Dividing through by \( \pi \):
\[ r_1^2 + r_2^2 = 74 \quad \text{(Equation 2)} \]

### Step 3: Substitute \( r_2 \) in terms of \( r_1 \)
From Equation 1, we can express \( r_2 \) in terms of \( r_1 \):
\[ r_2 = 12 - r_1 \]

### Step 4: Substitute \( r_2 \) into Equation 2
Now, substitute \( r_2 \) into Equation 2:
\[ r_1^2 + (12 - r_1)^2 = 74 \]

### Step 5: Expand and simplify the equation
Expanding \( (12 - r_1)^2 \):
\[ r_1^2 + (144 - 24r_1 + r_1^2) = 74 \]
Combining like terms:
\[ 2r_1^2 - 24r_1 + 144 = 74 \]
Subtracting 74 from both sides:
\[ 2r_1^2 - 24r_1 + 70 = 0 \]

### Step 6: Divide the equation by 2
To simplify, divide the entire equation by 2:
\[ r_1^2 - 12r_1 + 35 = 0 \]

### Step 7: Factor the quadratic equation
Now we will factor the quadratic:
\[ (r_1 - 7)(r_1 - 5) = 0 \]
Thus, we have two possible solutions for \( r_1 \):
\[ r_1 = 7 \quad \text{or} \quad r_1 = 5 \]

### Step 8: Find \( r_2 \)
Using Equation 1 to find \( r_2 \):
1. If \( r_1 = 7 \):
\[ r_2 = 12 - 7 = 5 \]
2. If \( r_1 = 5 \):
\[ r_2 = 12 - 5 = 7 \]

### Step 9: Calculate the diameters
The diameter \( d \) of a circle is given by \( d = 2r \):
1. For \( r_1 = 7 \) and \( r_2 = 5 \):
\[ d_1 = 2 \times 7 = 14 \, \text{cm}, \quad d_2 = 2 \times 5 = 10 \, \text{cm} \]

### Final Answer
The diameters of the circles are:
\[ \text{Diameter 1} = 14 \, \text{cm}, \quad \text{Diameter 2} = 10 \, \text{cm} \]

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