Equation of the circle which cuts the circle `x^2+y^2+2x+ 4y -4=0` and the lines `xy -2x -y+2=0` orthogonally, is

Options

A. `x^(2)+y^(2)-2x-4y-6=0`
B. `x^(2)+y^(2)-2x-4y+6=0`
C. `x^(2)+y^(2)-2x-4y-12=0`
D. none of these

Explanation

To find the equation of the circle that cuts the given circle and the line orthogonally, we can follow these steps:

### Step 1: Identify the given circle and line
The equation of the given circle is:
\[ x^2 + y^2 + 2x + 4y - 4 = 0 \]
We can rearrange this into standard form if needed, but we will use it as is for now.

The equation of the line is:
\[ xy - 2x - y + 2 = 0 \]
We can factor this to find its slope and intercepts.

### Step 2: Factor the line equation
Rearranging the line equation:
\[ xy - 2x - y + 2 = 0 \]
Factoring gives:
\[ (y - 2)(x - 1) = 0 \]
This means the line consists of two parts: \( x = 1 \) and \( y = 2 \).

### Step 3: Find the center of the required circle
Since the required circle cuts the given circle and the line orthogonally, we can assume the center of the required circle lies at the intersection of the line \( x = 1 \) and \( y = 2 \). Thus, the center of the required circle is:
\[ (1, 2) \]

### Step 4: Write the general equation of the circle
The general equation of a circle can be written as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Substituting the center coordinates \( (1, 2) \):
- The center gives us \( g = -1 \) and \( f = -2 \).

So, the equation becomes:
\[ x^2 + y^2 - 2x - 4y + c = 0 \]

### Step 5: Use the orthogonality condition
For two circles to intersect orthogonally, the condition is:
\[ 2g_1g_2 + f_1f_2 = c_1 + c_2 \]
Where:
- \( g_1, f_1, c_1 \) are from the first circle,
- \( g_2, f_2, c_2 \) are from the second circle.

From the given circle \( x^2 + y^2 + 2x + 4y - 4 = 0 \):
- \( g_1 = 1 \), \( f_1 = 2 \), \( c_1 = -4 \)

From our required circle:
- \( g_2 = -1 \), \( f_2 = -2 \), \( c_2 = c \)

Substituting into the orthogonality condition:
\[ 2(1)(-1) + (2)(-2) = -4 + c \]
This simplifies to:
\[ -2 - 4 = -4 + c \]
\[ -6 = -4 + c \]
Thus:
\[ c = -2 \]

### Step 6: Write the final equation of the circle
Substituting \( c = -6 \) back into the circle equation:
\[ x^2 + y^2 - 2x - 4y - 6 = 0 \]

### Final Answer
The required equation of the circle is:
\[ x^2 + y^2 - 2x - 4y - 6 = 0 \]