`x^(2)+y^(2)-6x+8y=-9`
The equation of the circle in xy-plane is shown above. What is the radius of the circle?

Options

A. `3`
B. `4`
C. `9`
D. `16`

Answer

To find the radius of the circle given by the equation \( x^{2} + y^{2} - 6x + 8y = -9 \), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation:
\[ x^{2} + y^{2} - 6x + 8y = -9 \]
To rewrite it in the standard form of a circle, we need to move \(-9\) to the left side:
\[ x^{2} + y^{2} - 6x + 8y + 9 = 0 \]

### Step 2: Identify coefficients for the general circle equation
The general equation of a circle is given by:
\[ x^{2} + y^{2} + 2gx + 2fy + c = 0 \]
From our rewritten equation, we can identify:
- The coefficient of \(x\) is \(-6\), which means \(2g = -6\) so \(g = -3\).
- The coefficient of \(y\) is \(8\), which means \(2f = 8\) so \(f = 4\).
- The constant term \(c\) is \(9\).

### Step 3: Calculate the radius
The radius \(r\) of the circle can be found using the formula:
\[ r = \sqrt{g^{2} + f^{2} - c} \]
Substituting the values we found:
- \(g = -3\)
- \(f = 4\)
- \(c = 9\)

We calculate:
\[ r = \sqrt{(-3)^{2} + (4)^{2} - 9} \]
Calculating each term:
\[ (-3)^{2} = 9 \quad \text{and} \quad (4)^{2} = 16 \]
So,
\[ r = \sqrt{9 + 16 - 9} \]
This simplifies to:
\[ r = \sqrt{16} = 4 \]

### Conclusion
The radius of the circle is \(4\).

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