If the angle between two equal circles with centres (3, 10) (-5, 4) is `120^@` then the radius of the circles is

Options

A. 10
B. 5
C. 2
D. 1

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Identify the centers of the circles
The centers of the two equal circles are given as:
- \( C_1 = (3, 10) \)
- \( C_2 = (-5, 4) \)

### Step 2: Calculate the distance \( D \) between the centers
The distance \( D \) between the two centers can be calculated using the distance formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of the centers:
\[ D = \sqrt{((-5) - 3)^2 + (4 - 10)^2} \]
\[ D = \sqrt{(-8)^2 + (-6)^2} \]
\[ D = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 3: Use the cosine rule for circles
Given that the angle between the two circles is \( 120^\circ \), we can use the cosine rule:
\[ \cos(120^\circ) = -\frac{1}{2} \]
The formula relating the angle between two circles and their radii is:
\[ \cos(\theta) = \frac{D^2 - R_1^2 - R_2^2}{2 R_1 R_2} \]
Since both circles are equal, we have \( R_1 = R_2 = R \):
\[ -\frac{1}{2} = \frac{D^2 - R^2 - R^2}{2R^2} \]
This simplifies to:
\[ -\frac{1}{2} = \frac{D^2 - 2R^2}{2R^2} \]

### Step 4: Substitute the value of \( D \)
Substituting \( D = 10 \):
\[ -\frac{1}{2} = \frac{10^2 - 2R^2}{2R^2} \]
\[ -\frac{1}{2} = \frac{100 - 2R^2}{2R^2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ -2R^2 = 100 - 2R^2 \]

### Step 6: Solve for \( R^2 \)
Adding \( 2R^2 \) to both sides:
\[ 0 = 100 \]
This indicates that we need to rearrange the equation correctly. Let's rewrite it:
\[ -2R^2 = 100 - 2R^2 \implies 0 = 100 \]
This is incorrect; we should have:
\[ -2R^2 + 2R^2 = 100 \implies 0 = 100 \]
This is a contradiction. Let's go back to the equation:
\[ -2R^2 = 100 - 2R^2 \implies 0 = 100 \]
This indicates we need to check our steps.

### Step 7: Correctly rearranging the equation
We should have:
\[ -2R^2 = 100 - 2R^2 \]
Adding \( 2R^2 \) to both sides:
\[ 0 = 100 \]
This is a contradiction. Let's go back to the equation:
\[ \frac{100 - 2R^2}{2R^2} = -\frac{1}{2} \]
Cross-multiplying gives:
\[ 100 - 2R^2 = -R^2 \]
Adding \( 2R^2 \) to both sides:
\[ 100 = R^2 \]
Taking the square root:
\[ R = 10 \]

### Final Answer
The radius of the circles is \( R = 10 \).