The eqaution of the circle concentric with `x^(2) + y^(2) - 3x + 4y + c = 0 ` and passing through the point (-1,-2) is

Options

A. A. `x^(2) + y^(2) - 3x + 4y + 2 = 0 `
B. B. ` x^(2) + y^(2) - 3x + 4y - 1 = 0 `
C. C. `x^(2) + y^(2) - 3x + 4y - 5 = 0 `
D. D. ` x^(2) + y^(2) - 3x + 4y = 0`

Answer

To find the equation of the circle that is concentric with the given circle \(x^2 + y^2 - 3x + 4y + c = 0\) and passes through the point \((-1, -2)\), we can follow these steps:

### Step 1: Identify the center of the given circle
The general form of the circle's equation is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Comparing this with the given equation \(x^2 + y^2 - 3x + 4y + c = 0\), we can identify:
- \(2g = -3 \Rightarrow g = -\frac{3}{2}\)
- \(2f = 4 \Rightarrow f = 2\)

Thus, the center of the given circle is:
\[ \left(-g, -f\right) = \left(\frac{3}{2}, -2\right) \]

### Step 2: Write the equation of the new circle
Since the new circle is concentric with the given circle, it will have the same center \(\left(\frac{3}{2}, -2\right)\). The equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting the center:
\[ \left(x - \frac{3}{2}\right)^2 + (y + 2)^2 = r^2 \]

### Step 3: Find the radius using the point \((-1, -2)\)
Since the circle passes through the point \((-1, -2)\), we can substitute this point into the equation to find \(r^2\):
\[ \left(-1 - \frac{3}{2}\right)^2 + (-2 + 2)^2 = r^2 \]
Calculating the left side:
\[ \left(-\frac{5}{2}\right)^2 + 0^2 = r^2 \]
\[ \frac{25}{4} = r^2 \]

### Step 4: Substitute \(r^2\) back into the equation
Now we substitute \(r^2\) back into the equation of the circle:
\[ \left(x - \frac{3}{2}\right)^2 + (y + 2)^2 = \frac{25}{4} \]

### Step 5: Expand the equation
Expanding the left side:
\[ \left(x^2 - 3x + \frac{9}{4}\right) + \left(y^2 + 4y + 4\right) = \frac{25}{4} \]
Combining terms:
\[ x^2 + y^2 - 3x + 4y + \frac{9}{4} + 4 = \frac{25}{4} \]
Converting \(4\) to quarters:
\[ 4 = \frac{16}{4} \]
Thus:
\[ x^2 + y^2 - 3x + 4y + \frac{9}{4} + \frac{16}{4} = \frac{25}{4} \]
This simplifies to:
\[ x^2 + y^2 - 3x + 4y + \frac{25}{4} = \frac{25}{4} \]
Subtracting \(\frac{25}{4}\) from both sides:
\[ x^2 + y^2 - 3x + 4y = 0 \]

### Final Answer
Thus, the equation of the circle is:
\[ x^2 + y^2 - 3x + 4y = 0 \]