If the point (2,-3) lies on the circle `x^(2) + y^(2) + 2 g x + 2fy + c = 0 ` which is concentric with the circle `x^(2) + y^(2) + 6x + 8y - 25 = 0` , then the value of c is

Options

A. (a) 1
B. (b) `-1`
C. (c) 49
D. (d) `-49`

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript:

### Step 1: Identify the equations of the circles
We have two circles:
1. Circle 1: \( x^2 + y^2 + 2gx + 2fy + c = 0 \) (Equation 1)
2. Circle 2: \( x^2 + y^2 + 6x + 8y - 25 = 0 \) (Equation 2)

### Step 2: Find the center of Circle 2
The general form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
The center of the circle can be found using the formula \( (-g, -f) \).

From Equation 2, we can compare coefficients:
- \( 2g = 6 \) → \( g = \frac{6}{2} = 3 \)
- \( 2f = 8 \) → \( f = \frac{8}{2} = 4 \)

Thus, the center of Circle 2 is:
\[ (-g, -f) = (-3, -4) \]

### Step 3: Since the circles are concentric
Since Circle 1 is concentric with Circle 2, it must have the same center:
\[ (-3, -4) \]

### Step 4: Substitute the point (2, -3) into Circle 1
The point (2, -3) lies on Circle 1, so we substitute \( x = 2 \) and \( y = -3 \) into Equation 1:
\[ 2^2 + (-3)^2 + 2(3)(2) + 2(4)(-3) + c = 0 \]

### Step 5: Simplify the equation
Calculating each term:
- \( 2^2 = 4 \)
- \( (-3)^2 = 9 \)
- \( 2(3)(2) = 12 \)
- \( 2(4)(-3) = -24 \)

Now substituting these values into the equation:
\[ 4 + 9 + 12 - 24 + c = 0 \]

### Step 6: Combine like terms
Combining the constants:
\[ 4 + 9 + 12 - 24 = 1 \]
Thus, we have:
\[ 1 + c = 0 \]

### Step 7: Solve for c
To find \( c \):
\[ c = -1 \]

### Conclusion
The value of \( c \) is:
\[ \boxed{-1} \]