Equation of a circle passing through origin is `x^(2) + y^(2) - 6x + 2 y = 0`. What is the equation of one of its diameter ?

Options

A. `x+3y = 0`
B. `x+y =0`
C. `x= y`
D. `3x+y = 0`

Explanation

To find the equation of one of the diameters of the circle given by the equation \( x^2 + y^2 - 6x + 2y = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
We start with the given equation of the circle:
\[ x^2 + y^2 - 6x + 2y = 0 \]
We can rearrange this equation to complete the square for both \( x \) and \( y \).

### Step 2: Completing the Square
1. For \( x^2 - 6x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]
2. For \( y^2 + 2y \):
\[ y^2 + 2y = (y + 1)^2 - 1 \]
Now, substituting these back into the equation gives:
\[ (x - 3)^2 - 9 + (y + 1)^2 - 1 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y + 1)^2 - 10 = 0 \]
or
\[ (x - 3)^2 + (y + 1)^2 = 10 \]
This shows that the circle is centered at \( (3, -1) \) with a radius of \( \sqrt{10} \).

### Step 3: Identify the Diameter's Endpoints
Since the circle passes through the origin \( (0, 0) \), we have one endpoint of the diameter at the origin. The other endpoint can be found by using the center of the circle and the radius.

The coordinates of the center are \( (3, -1) \). The diameter will pass through the center and the origin. The other endpoint of the diameter can be found by extending the line from the center through the origin.

### Step 4: Finding the Equation of the Diameter
The endpoints of the diameter are \( (0, 0) \) and \( (6, -2) \) (since the center is at \( (3, -1) \) and the distance to the origin is equal to the distance from the center to the other endpoint).

Using the two points \( (0, 0) \) and \( (6, -2) \), we can use the two-point form of the line equation:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]
Substituting the points:
\[ \frac{y - 0}{-2 - 0} = \frac{x - 0}{6 - 0} \]
This simplifies to:
\[ \frac{y}{-2} = \frac{x}{6} \]
Cross-multiplying gives:
\[ 6y + 2x = 0 \]
or
\[ 3y + x = 0 \]
Thus, the equation of one of the diameters of the circle is:
\[ x + 3y = 0 \]