The equation of the diameter of the circle `x^(2) + y^(2) - 6x + 2y = 0 ` which passes through origin is

Options

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

Answer

To find the equation of the diameter of the circle given by the equation \( x^2 + y^2 - 6x + 2y = 0 \) that passes through the origin, we can 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 match the standard form of a circle's equation.

### Step 2: Complete the Square
To rewrite the equation in standard form, we complete the square for the \(x\) and \(y\) terms.

1. For \(x\):
- The terms are \(x^2 - 6x\).
- Completing the square:
\[ x^2 - 6x = (x - 3)^2 - 9 \]

2. For \(y\):
- The terms are \(y^2 + 2y\).
- Completing the square:
\[ y^2 + 2y = (y + 1)^2 - 1 \]

Now substituting back into the equation:
\[ (x - 3)^2 - 9 + (y + 1)^2 - 1 = 0 \]
This simplifies to:
\[ (x - 3)^2 + (y + 1)^2 - 10 = 0 \]
Thus, we can rewrite it as:
\[ (x - 3)^2 + (y + 1)^2 = 10 \]

### Step 3: Identify the Center and Radius
From the standard form \((x - h)^2 + (y - k)^2 = r^2\), we identify:
- Center \(M(3, -1)\)
- Radius \(r = \sqrt{10}\)

### Step 4: Find the Equation of the Diameter
Since the diameter passes through the origin \((0, 0)\) and the center \(M(3, -1)\), we can find the slope of the line connecting these two points.

1. Slope \(m\) from the origin to the center \(M\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 0}{3 - 0} = \frac{-1}{3} \]

2. Using the point-slope form of a line, the equation of the line through the origin with this slope is:
\[ y - 0 = -\frac{1}{3}(x - 0) \]
Simplifying gives:
\[ y = -\frac{1}{3}x \]

### Step 5: Convert to Standard Form
To express this in standard form:
\[ y + \frac{1}{3}x = 0 \quad \text{or} \quad 3y + x = 0 \]
Thus, the equation of the diameter is:
\[ y + 3x = 0 \]

### Final Answer
The equation of the diameter of the circle that passes through the origin is:
\[ \boxed{y + 3x = 0} \]