`x ^(2) + y^(2) + 8x -20y =28`
What is the diameter of the circle given by the equation above ?

Options

A. 12
B. 24
C. 28
D. 56

Answer

To find the diameter of the circle given by the equation \( x^2 + y^2 + 8x - 20y = 28 \), we will first convert the equation into the standard form of a circle. The standard form of a circle is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \( (h, k) \) is the center of the circle and \( r \) is the radius.

### Step 1: Rearranging the equation
Start with the given equation:

\[ x^2 + y^2 + 8x - 20y = 28 \]

### Step 2: Completing the square for \( x \)
To complete the square for the \( x \) terms, we take the coefficient of \( x \) (which is 8), halve it to get 4, and then square it to get 16. We will add and subtract 16 in the equation:

\[ x^2 + 8x + 16 - 16 \]

This simplifies to:

\[ (x + 4)^2 - 16 \]

### Step 3: Completing the square for \( y \)
Next, for the \( y \) terms, take the coefficient of \( y \) (which is -20), halve it to get -10, and then square it to get 100. We will add and subtract 100 in the equation:

\[ y^2 - 20y + 100 - 100 \]

This simplifies to:

\[ (y - 10)^2 - 100 \]

### Step 4: Substitute back into the equation
Now, substitute the completed squares back into the equation:

\[ (x + 4)^2 - 16 + (y - 10)^2 - 100 = 28 \]

Combine the constants on the left side:

\[ (x + 4)^2 + (y - 10)^2 - 116 = 28 \]

Adding 116 to both sides gives:

\[ (x + 4)^2 + (y - 10)^2 = 144 \]

### Step 5: Identify the radius
Now, we can see that this is in the standard form of a circle:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Here, \( r^2 = 144 \). Thus, the radius \( r \) is:

\[ r = \sqrt{144} = 12 \]

### Step 6: Calculate the diameter
The diameter \( D \) of the circle is twice the radius:

\[ D = 2r = 2 \times 12 = 24 \]

### Final Answer
Thus, the diameter of the circle is:

\[ \boxed{24} \]