Find the equation of the circle passing through the points of intersection of the circles `x^2 + y^2 - 2x - 4y - 4 = 0` and `x^2 + y^2 - 10x - 12y +40 = 0` and whose radius is 4.

Options

A. `x^(2)+y^(2) -2x-15 =0`
B. `x^(2) +y^(2) _2y - 15 =0`
C. `x^(2) +y^(2)-2x+2y -15 =0`
D. `x^(2) +y^(2) +2x+2y -15=0`

Answer

To find the equation of the circle passing through the points of intersection of the two given circles and having a radius of 4, we can follow these steps:

### Step 1: Identify the equations of the circles
The equations of the circles are:
1. \( S_1: x^2 + y^2 - 2x - 4y - 4 = 0 \)
2. \( S_2: x^2 + y^2 - 10x - 12y + 40 = 0 \)

### Step 2: Use the formula for the equation of the circle through the intersection of two circles
The general equation of a circle passing through the intersection of two circles \( S_1 \) and \( S_2 \) can be expressed as:
\[ S_1 + \lambda S_2 = 0 \]
where \( \lambda \) is a parameter.

### Step 3: Substitute the equations into the formula
Substituting the equations of \( S_1 \) and \( S_2 \):
\[ (x^2 + y^2 - 2x - 4y - 4) + \lambda (x^2 + y^2 - 10x - 12y + 40) = 0 \]
This simplifies to:
\[ (1 + \lambda)x^2 + (1 + \lambda)y^2 + (-2 - 10\lambda)x + (-4 - 12\lambda)y + (-4 + 40\lambda) = 0 \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ (1 + \lambda)x^2 + (1 + \lambda)y^2 + (-2 - 10\lambda)x + (-4 - 12\lambda)y + (36\lambda - 4) = 0 \]

### Step 5: Identify coefficients for the general circle equation
The general form of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our equation, we can identify:
- \( g = \frac{-(-2 - 10\lambda)}{2(1 + \lambda)} = \frac{2 + 10\lambda}{2(1 + \lambda)} \)
- \( f = \frac{-(-4 - 12\lambda)}{2(1 + \lambda)} = \frac{4 + 12\lambda}{2(1 + \lambda)} \)
- \( c = \frac{36\lambda - 4}{1 + \lambda} \)

### Step 6: Use the radius condition
The radius \( r \) of the circle is given by:
\[ r^2 = g^2 + f^2 - c \]
We know \( r = 4 \), so \( r^2 = 16 \).

### Step 7: Substitute \( g \), \( f \), and \( c \) into the radius equation
Substituting the expressions for \( g \), \( f \), and \( c \) into the equation:
\[ 16 = \left(\frac{2 + 10\lambda}{2(1 + \lambda)}\right)^2 + \left(\frac{4 + 12\lambda}{2(1 + \lambda)}\right)^2 - \frac{36\lambda - 4}{1 + \lambda} \]

### Step 8: Solve for \( \lambda \)
This equation can be simplified and solved for \( \lambda \). After solving, we find two possible values for \( \lambda \):
1. \( \lambda = 7 \)
2. \( \lambda = -\frac{1}{5} \)

### Step 9: Substitute \( \lambda \) back to find the equations of the circles
1. For \( \lambda = 7 \):
\[ S = 2x^2 + 2y^2 - 18x - 22y + 69 = 0 \]
2. For \( \lambda = -\frac{1}{5} \):
\[ S = x^2 + y^2 - 2y - 15 = 0 \]

### Final Answer
The equations of the circles passing through the intersection points are:
1. \( 2x^2 + 2y^2 - 18x - 22y + 69 = 0 \)
2. \( x^2 + y^2 - 2y - 15 = 0 \)