For the circle `x ^(2) +y^(2) + 3x +3y =0,` which of the following relation is true ?

Options

A. Centre lies on X-axis
B. Centre lies on Y-axis
C. Centre is at origin
D. Circle passes through origin

Answer

To solve the problem, we need to analyze the given equation of the circle and determine the properties of the circle based on its center and whether it passes through the origin.

### Step-by-Step Solution:

1. **Write the given equation of the circle**:
\[ x^2 + y^2 + 3x + 3y = 0 \]

2. **Compare with the general equation of a circle**:
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
Here, we can identify:
- \(2g = 3\)
- \(2f = 3\)
- \(c = 0\)

3. **Find the values of \(g\) and \(f\)**:
From the equations:
\[ 2g = 3 \implies g = \frac{3}{2} \]
\[ 2f = 3 \implies f = \frac{3}{2} \]

4. **Determine the center of the circle**:
The center of the circle is given by the coordinates \((-g, -f)\):
\[ \text{Center} = \left(-\frac{3}{2}, -\frac{3}{2}\right) \]

5. **Check if the center lies on the x-axis or y-axis**:
- For the center to lie on the x-axis, the y-coordinate must be 0. Here, it is \(-\frac{3}{2}\), so it does not lie on the x-axis.
- For the center to lie on the y-axis, the x-coordinate must be 0. Here, it is \(-\frac{3}{2}\), so it does not lie on the y-axis.

6. **Check if the center is at the origin**:
The center is \(\left(-\frac{3}{2}, -\frac{3}{2}\right)\), which is not the origin \((0, 0)\).

7. **Check if the circle passes through the origin**:
The circle passes through the origin if \(c = 0\). In our case, since the equation can be rewritten as:
\[ x^2 + y^2 + 3x + 3y = 0 \]
and \(c = 0\), we conclude that the circle does indeed pass through the origin.

### Conclusion:
The correct relation from the given options is:
- The circle passes through the origin.