A circle `x ^(2) + y^(2) + 2gx + 2fy + c=0` passing through`(4,-2) ` is concentric to the circle `x ^(2) + y ^(2) -2x + 4y + 20 =0,` then the value of c will be

Options

A. `-4`
B. 4
C. 0
D. 1

Answer

To solve the problem, we need to find the value of \( c \) for the circle given by the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) that is concentric to the circle \( x^2 + y^2 - 2x + 4y + 20 = 0 \) and passes through the point \( (4, -2) \).

### Step 1: Identify the center of the given circle
The given circle is:
\[ x^2 + y^2 - 2x + 4y + 20 = 0 \]
To find the center, we can rewrite this in the standard form of a circle:
\[ (x^2 - 2x) + (y^2 + 4y) + 20 = 0 \]
Completing the square for \( x \) and \( y \):
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 + 20 = 0 \]
\[ (x - 1)^2 + (y + 2)^2 + 15 = 0 \]
Thus, the center of the given circle is \( (1, -2) \).

### Step 2: Write the equation of the concentric circle
Since the circles are concentric, the center of the circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \) must also be \( (1, -2) \). Therefore, we can express the equation of the concentric circle as:
\[ (x - 1)^2 + (y + 2)^2 + \lambda = 0 \]
Expanding this gives:
\[ x^2 - 2x + 1 + y^2 + 4y + 4 + \lambda = 0 \]
This simplifies to:
\[ x^2 + y^2 - 2x + 4y + (5 + \lambda) = 0 \]
From this, we can identify that:
\[ 2g = -2 \quad \text{and} \quad 2f = 4 \quad \text{and} \quad c = 5 + \lambda \]

### Step 3: Substitute the point (4, -2)
Since the circle passes through the point \( (4, -2) \), we substitute \( x = 4 \) and \( y = -2 \) into the equation:
\[ (4 - 1)^2 + (-2 + 2)^2 + \lambda = 0 \]
Calculating this:
\[ 3^2 + 0^2 + \lambda = 0 \]
\[ 9 + \lambda = 0 \]
Thus, we find:
\[ \lambda = -9 \]

### Step 4: Find the value of \( c \)
Now substituting \( \lambda \) back into the equation for \( c \):
\[ c = 5 + \lambda = 5 - 9 = -4 \]

### Conclusion
The value of \( c \) is:
\[ \boxed{-4} \]