The equation of the circle concentric with `x^(2)+y^(2)-6x+4y-3=0` and having radius 5 is

Options

A. `x^(2)+y^(2)-6x+4y-12=0`
B. `x^(2)+y^(2)-2x+8y-33=0`
C. `x^(2)+y^(2)+6x-4y-12=0`
D. `x^(2)+y^(2)+x+8y+33=0`

Answer

To find the equation of a circle that is concentric with the given circle \(x^2 + y^2 - 6x + 4y - 3 = 0\) and has a radius of 5, we can follow these steps:

### Step 1: Identify the center of the given circle
The general form of a circle's equation is given by:
\[ x^2 + y^2 + Dx + Ey + F = 0 \]
where the center \((h, k)\) can be found using the formulas:
\[ h = -\frac{D}{2}, \quad k = -\frac{E}{2} \]
In our case, \(D = -6\) and \(E = 4\).

Calculating the center:
\[ h = -\frac{-6}{2} = 3, \quad k = -\frac{4}{2} = -2 \]
Thus, the center of the given circle is \((3, -2)\).

### 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 \((3, -2)\) but a different radius. The radius of the new circle is given as 5.

The equation of a circle with center \((h, k)\) and radius \(r\) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \(h = 3\), \(k = -2\), and \(r = 5\):
\[ (x - 3)^2 + (y + 2)^2 = 5^2 \]
This simplifies to:
\[ (x - 3)^2 + (y + 2)^2 = 25 \]

### Step 3: Expand the equation
Now we will expand the equation:
\[ (x - 3)^2 = x^2 - 6x + 9 \]
\[ (y + 2)^2 = y^2 + 4y + 4 \]
Adding these together:
\[ x^2 - 6x + 9 + y^2 + 4y + 4 = 25 \]
Combining like terms:
\[ x^2 + y^2 - 6x + 4y + 13 = 25 \]
Now, rearranging gives:
\[ x^2 + y^2 - 6x + 4y + 13 - 25 = 0 \]
Thus:
\[ x^2 + y^2 - 6x + 4y - 12 = 0 \]

### Final Answer
The equation of the circle concentric with the given circle and having a radius of 5 is:
\[ x^2 + y^2 - 6x + 4y - 12 = 0 \]