Equation of the circle concentric with the circle
` x^(2) + y^(2) + 8x + 10 y - 7 = 0 ` , and passing through the centre of the circle ` x^(2) + y^(2) - 4x - 6y = 0 `,is

Options

A. `x^(2)+y^(2)+8x+10y +59=0`
B. `x^(2)+y^(2)+8x+10y-59=0`
C. `x^(2)+y^(2)-4x-6y+87=0`
D. `x^(2)+y^(2)-4x-6y-87=0`

Explanation

To find the equation of the circle that is concentric with the given circle and passes through the center of another circle, we will follow these steps:

### Step 1: Identify the center of the second circle
The equation of the second circle is given by:
\[ x^2 + y^2 - 4x - 6y = 0 \]

To find the center, we can rewrite this equation in standard form. We complete the square for both \(x\) and \(y\).

1. Rearranging:
\[ x^2 - 4x + y^2 - 6y = 0 \]

2. Completing the square:
- For \(x^2 - 4x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]
- For \(y^2 - 6y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]

3. Substituting back:
\[ (x - 2)^2 - 4 + (y - 3)^2 - 9 = 0 \]
\[ (x - 2)^2 + (y - 3)^2 - 13 = 0 \]
\[ (x - 2)^2 + (y - 3)^2 = 13 \]

Thus, the center of the second circle is at the point \( (2, 3) \).

### Step 2: Identify the equation of the first circle
The first circle is given by:
\[ x^2 + y^2 + 8x + 10y - 7 = 0 \]

We can also rewrite this in standard form to find its center and radius.

1. Rearranging:
\[ x^2 + 8x + y^2 + 10y = 7 \]

2. Completing the square:
- For \(x^2 + 8x\):
\[ x^2 + 8x = (x + 4)^2 - 16 \]
- For \(y^2 + 10y\):
\[ y^2 + 10y = (y + 5)^2 - 25 \]

3. Substituting back:
\[ (x + 4)^2 - 16 + (y + 5)^2 - 25 = 7 \]
\[ (x + 4)^2 + (y + 5)^2 - 41 = 7 \]
\[ (x + 4)^2 + (y + 5)^2 = 48 \]

The center of this circle is at \( (-4, -5) \).

### Step 3: Write the equation of the concentric circle
Since we want a circle that is concentric with the first circle, it will have the same center as the first circle, which is \( (-4, -5) \).

The general form of the equation of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( (h, k) \) is the center and \( r \) is the radius.

The equation of the concentric circle can be written as:
\[ (x + 4)^2 + (y + 5)^2 = r^2 \]

### Step 4: Determine the radius using the point (2, 3)
Since the circle passes through the point \( (2, 3) \), we substitute \( x = 2 \) and \( y = 3 \) into the equation:
\[ (2 + 4)^2 + (3 + 5)^2 = r^2 \]
\[ 6^2 + 8^2 = r^2 \]
\[ 36 + 64 = r^2 \]
\[ r^2 = 100 \]

### Step 5: Write the final equation
Now we can substitute \( r^2 = 100 \) back into the equation of the circle:
\[ (x + 4)^2 + (y + 5)^2 = 100 \]

Expanding this gives:
\[ x^2 + 8x + 16 + y^2 + 10y + 25 = 100 \]
\[ x^2 + y^2 + 8x + 10y + 41 - 100 = 0 \]
\[ x^2 + y^2 + 8x + 10y - 59 = 0 \]

Thus, the equation of the circle is:
\[ x^2 + y^2 + 8x + 10y - 59 = 0 \]

### Final Answer
The equation of the required circle is:
\[ x^2 + y^2 + 8x + 10y - 59 = 0 \]