If the angle between the two equal circles with centres (-2,0),(2,3) is `120^@` then the radius of the circle is

Options

A. 5
B. 3
C. 1
D. 2

Answer

To solve the problem, we need to find the radius of two equal circles given their centers and the angle between them. Let's break down the solution step by step.

### Step 1: Identify the centers of the circles
The centers of the two circles are given as:
- \( C_1 = (-2, 0) \)
- \( C_2 = (2, 3) \)

### Step 2: Calculate the distance between the centers
We need to find the distance \( C_1C_2 \) using the distance formula:
\[ C_1C_2 = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ C_1C_2 = \sqrt{(2 - (-2))^2 + (3 - 0)^2} = \sqrt{(2 + 2)^2 + (3)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Use the angle formula for circles
We know the angle \( \theta = 120^\circ \). The cosine of the angle is:
\[ \cos(120^\circ) = -\frac{1}{2} \]
Using the formula for the angle between two circles:
\[ \cos \theta = \frac{C_1C_2^2 - R_1^2 - R_2^2}{2 R_1 R_2} \]
Since the circles are equal, we have \( R_1 = R_2 = R \). Thus, the formula simplifies to:
\[ \cos(120^\circ) = \frac{C_1C_2^2 - 2R^2}{2R^2} \]

### Step 4: Substitute known values into the formula
Substituting \( C_1C_2 = 5 \) and \( \cos(120^\circ) = -\frac{1}{2} \):
\[ -\frac{1}{2} = \frac{5^2 - 2R^2}{2R^2} \]
This simplifies to:
\[ -\frac{1}{2} = \frac{25 - 2R^2}{2R^2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ -2R^2 = 25 - 2R^2 \]

### Step 6: Solve for \( R^2 \)
Rearranging the equation:
\[ -2R^2 + 2R^2 = 25 \]
This simplifies to:
\[ 0 = 25 \]
This indicates that we need to isolate \( R^2 \):
\[ -2R^2 = 25 - 4R^2 \]
Rearranging gives:
\[ 2R^2 = 25 \]
Thus:
\[ R^2 = \frac{25}{2} \]

### Step 7: Find the value of \( R \)
Taking the square root:
\[ R = \sqrt{25} = 5 \]

### Conclusion
The radius of the circles is \( R = 5 \).