The point (3,-4) lies on both the circles `x^2+y^2-2x+8y+13=0` and `x^2+y^2-4x+6y+11=0` . Then the angle between the circles is

Options

A. `60^@`
B. `30^@`
C. `120^@`
D. `135^@`

Answer

To find the angle between the two circles given by the equations \(x^2 + y^2 - 2x + 8y + 13 = 0\) and \(x^2 + y^2 - 4x + 6y + 11 = 0\), we will follow these steps:

### Step 1: Rewrite the equations of the circles in standard form
The general form of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

#### For the first circle:
Starting with the equation:
\[ x^2 + y^2 - 2x + 8y + 13 = 0 \]
We can rearrange it:
\[ x^2 - 2x + y^2 + 8y + 13 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 2x + 1) + (y^2 + 8y + 16) = -13 + 1 + 16 \]
This simplifies to:
\[ (x - 1)^2 + (y + 4)^2 = 2 \]
Thus, the center \(C_1\) is \((1, -4)\) and the radius \(R_1 = \sqrt{2}\).

#### For the second circle:
Starting with the equation:
\[ x^2 + y^2 - 4x + 6y + 11 = 0 \]
Rearranging gives:
\[ x^2 - 4x + y^2 + 6y + 11 = 0 \]
Completing the square:
\[ (x^2 - 4x + 4) + (y^2 + 6y + 9) = -11 + 4 + 9 \]
This simplifies to:
\[ (x - 2)^2 + (y + 3)^2 = 2 \]
Thus, the center \(C_2\) is \((2, -3)\) and the radius \(R_2 = \sqrt{2}\).

### Step 2: Find the distance between the centers of the circles
The distance \(d\) between the centers \(C_1(1, -4)\) and \(C_2(2, -3)\) is calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(2 - 1)^2 + (-3 + 4)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Use the formula for the angle between two circles
The angle \(\theta\) between two circles can be found using the formula:
\[ \cos \theta = \frac{d^2 - R_1^2 - R_2^2}{2 R_1 R_2} \]
Substituting the values:
\[ \cos \theta = \frac{(\sqrt{2})^2 - (\sqrt{2})^2 - (\sqrt{2})^2}{2 \cdot \sqrt{2} \cdot \sqrt{2}} = \frac{2 - 2 - 2}{4} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 4: Find the angle \(\theta\)
Now, we find \(\theta\) using:
\[ \theta = \cos^{-1}(-\frac{1}{2}) = 120^\circ \]

### Conclusion
The angle between the two circles is \(120^\circ\).