The radius of the circle `3x^(2)+3y^(2)+9x+8y-4=0` is

Options

A. `(sqrt(193))/(3)` units
B. `(sqrt(193))/(6)` units
C. `(sqrt(129))/(3)` units
D. none

Answer

To find the radius of the circle given by the equation \(3x^2 + 3y^2 + 9x + 8y - 4 = 0\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
First, we divide the entire equation by 3 to simplify it:

\[ x^2 + y^2 + 3x + \frac{8}{3}y - \frac{4}{3} = 0 \]

### Step 2: Identify coefficients
Now we can compare this with the standard form of the circle's equation:

\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]

From our equation, we can identify:
- \(2g = 3\) (so \(g = \frac{3}{2}\))
- \(2f = \frac{8}{3}\) (so \(f = \frac{4}{3}\))
- \(c = -\frac{4}{3}\)

### Step 3: Calculate the radius
The radius \(r\) of the circle can be calculated using the formula:

\[ r = \sqrt{g^2 + f^2 - c} \]

Substituting the values of \(g\), \(f\), and \(c\):

\[ g^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
\[ f^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
\[ -c = -\left(-\frac{4}{3}\right) = \frac{4}{3} \]

### Step 4: Find a common denominator
Now, we need to find a common denominator to combine these fractions. The least common multiple of 4, 9, and 3 is 36.

Convert each term:
- \(g^2 = \frac{9}{4} = \frac{81}{36}\)
- \(f^2 = \frac{16}{9} = \frac{64}{36}\)
- \(-c = \frac{4}{3} = \frac{48}{36}\)

### Step 5: Combine the fractions
Now we can combine these:

\[ g^2 + f^2 - c = \frac{81}{36} + \frac{64}{36} + \frac{48}{36} = \frac{193}{36} \]

### Step 6: Calculate the radius
Now we can find the radius:

\[ r = \sqrt{\frac{193}{36}} = \frac{\sqrt{193}}{6} \]

### Conclusion
Thus, the radius of the circle is:

\[ \frac{\sqrt{193}}{6} \]