The equation of the circle which passes through the point (1, 1) and cuts the circles `x^2+y^2-8x-2y+16=0` and `x^2+y^2-4x-4y-1=0` orthogonally is

Options

A. `x^(2)+y^(2)+16x+12y+2=0`
B. `x^(2)+y^(2)-16x-12y-2=0`
C. `3(x^2+y^2)-14x+23y-15=0`
D. none

Answer

To find the equation of the circle that passes through the point (1, 1) and cuts the given circles orthogonally, we can follow these steps:

### Step 1: Write the general equation of the circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \(g\), \(f\), and \(c\) are constants to be determined.

### Step 2: Use the point (1, 1)
Since the circle passes through the point (1, 1), we can substitute \(x = 1\) and \(y = 1\) into the equation:
\[ 1^2 + 1^2 + 2g(1) + 2f(1) + c = 0 \]
This simplifies to:
\[ 2 + 2g + 2f + c = 0 \]
Rearranging gives us:
\[ 2g + 2f + c = -2 \quad \text{(Equation 1)} \]

### Step 3: Analyze the first given circle
The first circle is given by:
\[ x^2 + y^2 - 8x - 2y + 16 = 0 \]
We can rewrite it in standard form:
\[ (x - 4)^2 + (y - 1)^2 = 1 \]
From this, we can identify:
- \(g_1 = -4\)
- \(f_1 = -1\)
- \(c_1 = -16\)

### Step 4: Orthogonality condition with the first circle
The condition for two circles to cut orthogonally is:
\[ 2g_1g + 2f_1f = c_1 + c_2 \]
Substituting the values:
\[ 2(-4)g + 2(-1)f = -16 + c \]
This simplifies to:
\[ -8g - 2f = -16 + c \quad \text{(Equation 2)} \]

### Step 5: Analyze the second given circle
The second circle is given by:
\[ x^2 + y^2 - 4x - 4y - 1 = 0 \]
Rewriting it gives:
\[ (x - 2)^2 + (y - 2)^2 = 5 \]
From this, we identify:
- \(g_2 = -2\)
- \(f_2 = -2\)
- \(c_2 = 1\)

### Step 6: Orthogonality condition with the second circle
Using the orthogonality condition again:
\[ 2g_2g + 2f_2f = c_2 + c \]
Substituting the values:
\[ 2(-2)g + 2(-2)f = 1 + c \]
This simplifies to:
\[ -4g - 4f = 1 + c \quad \text{(Equation 3)} \]

### Step 7: Solve the system of equations
Now we have three equations:
1. \(2g + 2f + c = -2\) (Equation 1)
2. \(-8g - 2f = -16 + c\) (Equation 2)
3. \(-4g - 4f = 1 + c\) (Equation 3)

We can solve these equations step by step.

### Step 8: Solve Equation 1 and Equation 2
From Equation 1:
\[ c = -2 - 2g - 2f \]
Substituting \(c\) into Equation 2:
\[ -8g - 2f = -16 + (-2 - 2g - 2f) \]
This simplifies to:
\[ -8g - 2f = -18 - 2g - 2f \]
Rearranging gives:
\[ -6g = -18 \quad \Rightarrow \quad g = 3 \]

### Step 9: Substitute \(g\) back to find \(f\) and \(c\)
Substituting \(g = 3\) into Equation 1:
\[ 2(3) + 2f + c = -2 \quad \Rightarrow \quad 6 + 2f + c = -2 \]
Thus:
\[ 2f + c = -8 \quad \text{(Equation 4)} \]
Now substitute \(g = 3\) into Equation 3:
\[ -4(3) - 4f = 1 + c \quad \Rightarrow \quad -12 - 4f = 1 + c \]
Rearranging gives:
\[ c = -13 - 4f \quad \text{(Equation 5)} \]

### Step 10: Solve Equations 4 and 5
Substituting Equation 5 into Equation 4:
\[ 2f + (-13 - 4f) = -8 \]
This simplifies to:
\[ -2f - 13 = -8 \quad \Rightarrow \quad -2f = 5 \quad \Rightarrow \quad f = -\frac{5}{2} \]

### Step 11: Find \(c\)
Substituting \(f = -\frac{5}{2}\) into Equation 5:
\[ c = -13 - 4(-\frac{5}{2}) = -13 + 10 = -3 \]

### Step 12: Write the final equation of the circle
Now we have:
- \(g = 3\)
- \(f = -\frac{5}{2}\)
- \(c = -3\)

The equation of the required circle is:
\[ x^2 + y^2 + 6x - 5y - 3 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 + 6x - 5y - 3 = 0 \]