If (-3,2) lies on the circle `x^(2)+y^(2) +2g x +2fy+c=0`, which is concentric with the circle `x^(2)+y^(2) +6x + 8y - 5 =0`, then c is

Options

A. 11
B. `-11`
C. 24
D. none

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Identify the equations of the circles
We have two circles:
1. Circle 1: \( x^2 + y^2 + 2gx + 2fy + c = 0 \)
2. Circle 2: \( x^2 + y^2 + 6x + 8y - 5 = 0 \)

### Step 2: Determine 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 values of \( g \) and \( f \):
- The center coordinates are \( (-g, -f) \).

From Circle 2, we can extract the coefficients:
- \( 2g = 6 \) implies \( g = 3 \)
- \( 2f = 8 \) implies \( f = 4 \)

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

### Step 3: Concentric circles
Since the circles are concentric, Circle 1 must also have the same center:
- Therefore, the center of Circle 1 is also \( (-3, -4) \).

### Step 4: Substitute the point into Circle 1's equation
We know that the point \( (-3, 2) \) lies on Circle 1. We will substitute \( x = -3 \) and \( y = 2 \) into the equation of Circle 1:
\[ (-3)^2 + (2)^2 + 2g(-3) + 2f(2) + c = 0 \]

### Step 5: Calculate the left-hand side
Calculating each term:
- \( (-3)^2 = 9 \)
- \( (2)^2 = 4 \)
- \( 2g(-3) = 2(3)(-3) = -18 \)
- \( 2f(2) = 2(4)(2) = 16 \)

Putting it all together:
\[ 9 + 4 - 18 + 16 + c = 0 \]

### Step 6: Simplify the equation
Combine the constants:
\[ 9 + 4 - 18 + 16 = 11 \]
So, we have:
\[ 11 + c = 0 \]

### Step 7: Solve for \( c \)
To find \( c \):
\[ c = -11 \]

### Final Answer
Thus, the value of \( c \) is:
\[ \boxed{-11} \]