I. The coordinate of the center are (2, -3).
II. The coordinate of the center are (-2, 3).
III. The length of the radius is `5sqrt(2)`.
IV. The length of the radius is 50.
Q. If an equation of a circle is `x^(2)+4y+y^(2)-6y=37`, which of the statements above are true?

Options

A. I and III
B. I and IV
C. II and III
D. II and IV

Answer

To determine which statements about the circle are true, we need to analyze the given equation of the circle:

\[ x^2 + 4x + y^2 - 6y = 37 \]

### Step 1: Rearranging the Equation
We will rearrange the equation to bring it into the standard form of a circle, which is:

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

To do this, we will complete the square for both \(x\) and \(y\).

### Step 2: Completing the Square for \(x\)
The \(x\) terms are \(x^2 + 4x\). To complete the square:

1. Take half of the coefficient of \(x\) (which is 4), square it:
\[ \left(\frac{4}{2}\right)^2 = 4 \]
2. Add and subtract this square inside the equation.

So, we rewrite:

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

### Step 3: Completing the Square for \(y\)
The \(y\) terms are \(y^2 - 6y\). To complete the square:

1. Take half of the coefficient of \(y\) (which is -6), square it:
\[ \left(\frac{-6}{2}\right)^2 = 9 \]
2. Add and subtract this square inside the equation.

So, we rewrite:

\[ y^2 - 6y = (y^2 - 6y + 9) - 9 = (y - 3)^2 - 9 \]

### Step 4: Substitute Back into the Equation
Now substituting back into the original equation:

\[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 37 \]

This simplifies to:

\[ (x + 2)^2 + (y - 3)^2 - 13 = 37 \]

Adding 13 to both sides gives:

\[ (x + 2)^2 + (y - 3)^2 = 50 \]

### Step 5: Identify the Center and Radius
From the equation \((x + 2)^2 + (y - 3)^2 = 50\):

- The center \((h, k)\) is \((-2, 3)\).
- The radius \(r\) is \(\sqrt{50} = 5\sqrt{2}\).

### Conclusion
Now we can evaluate the statements:

1. The coordinates of the center are (2, -3) - **False**
2. The coordinates of the center are (-2, 3) - **True**
3. The length of the radius is \(5\sqrt{2}\) - **True**
4. The length of the radius is 50 - **False**

### Final Answer
The true statements are:
- Statement II: The coordinates of the center are (-2, 3).
- Statement III: The length of the radius is \(5\sqrt{2}\).