The sum of the areas of two circles which touch each other externally is `153pi`. lf the sum of their radii is 15, find the ratio of the larger to the smaller radius

Options

A. 4
B. 2
C. 3
D. None of these

Explanation

To solve the problem step by step, we will use the given information about the areas of the circles and their radii.

### Step 1: Set Up the Equations
Let the radii of the two circles be \( R_1 \) and \( R_2 \). According to the problem, we know:

1. The sum of the areas of the two circles is \( 153\pi \).
\[ \pi R_1^2 + \pi R_2^2 = 153\pi \]
Dividing both sides by \( \pi \):
\[ R_1^2 + R_2^2 = 153 \]

2. The sum of the radii is \( 15 \):
\[ R_1 + R_2 = 15 \]

### Step 2: Use the Identity
We can use the identity:
\[ (R_1 + R_2)^2 = R_1^2 + R_2^2 + 2R_1R_2 \]
Substituting the known values:
\[ 15^2 = R_1^2 + R_2^2 + 2R_1R_2 \]
Calculating \( 15^2 \):
\[ 225 = R_1^2 + R_2^2 + 2R_1R_2 \]

### Step 3: Substitute the Known Value
Now we can substitute \( R_1^2 + R_2^2 = 153 \) into the equation:
\[ 225 = 153 + 2R_1R_2 \]
Subtract \( 153 \) from both sides:
\[ 225 - 153 = 2R_1R_2 \]
Calculating the left side:
\[ 72 = 2R_1R_2 \]
Dividing both sides by \( 2 \):
\[ R_1R_2 = 36 \]

### Step 4: Set Up the Quadratic Equation
Now we have two equations:
1. \( R_1 + R_2 = 15 \)
2. \( R_1R_2 = 36 \)

We can use these to form a quadratic equation. Let \( R_1 \) and \( R_2 \) be the roots of the equation:
\[ x^2 - (R_1 + R_2)x + R_1R_2 = 0 \]
Substituting the known values:
\[ x^2 - 15x + 36 = 0 \]

### Step 5: Solve the Quadratic Equation
Now we will solve the quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -15, c = 36 \):
\[ x = \frac{15 \pm \sqrt{(-15)^2 - 4 \cdot 1 \cdot 36}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{15 \pm \sqrt{225 - 144}}{2} \]
\[ x = \frac{15 \pm \sqrt{81}}{2} \]
\[ x = \frac{15 \pm 9}{2} \]
Calculating the two possible values:
1. \( x = \frac{24}{2} = 12 \)
2. \( x = \frac{6}{2} = 3 \)

### Step 6: Find the Ratio of the Radii
Thus, the radii are \( R_1 = 12 \) and \( R_2 = 3 \). The ratio of the larger radius to the smaller radius is:
\[ \text{Ratio} = \frac{R_1}{R_2} = \frac{12}{3} = 4 \]

### Final Answer
The ratio of the larger radius to the smaller radius is \( 4:1 \).

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