Equation of the diameter of the circle `x^(2)+y^(2)-2x+4y=0` which passes through the origin is

Options

A. `x+2y=0`
B. `x-2y=0`
C. `2x+y=0`
D. `2x-y=0`

Answer

To find the equation of the diameter of the circle given by the equation \( x^2 + y^2 - 2x + 4y = 0 \) that passes through the origin, we can follow these steps:

### Step 1: Rewrite the Circle Equation in Standard Form
We start with the equation of the circle:
\[ x^2 + y^2 - 2x + 4y = 0 \]
To rewrite it in standard form, we complete the square for both \(x\) and \(y\).

### Step 2: Completing the Square
1. For \(x\):
\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]
2. For \(y\):
\[ y^2 + 4y \rightarrow (y + 2)^2 - 4 \]

Putting it all together:
\[ (x - 1)^2 - 1 + (y + 2)^2 - 4 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y + 2)^2 = 5 \]
This shows that the center of the circle is at \( (1, -2) \) and the radius is \( \sqrt{5} \).

### Step 3: Finding the Diameter
Since the diameter passes through the origin \( (0, 0) \) and the center of the circle is \( (1, -2) \), we can find the coordinates of the other end of the diameter.

Let the other end of the diameter be \( (a, b) \). The midpoint of the diameter (which is the center of the circle) is given by:
\[ \left( \frac{0 + a}{2}, \frac{0 + b}{2} \right) = (1, -2) \]

### Step 4: Setting Up the Equations
From the midpoint formula, we have:
1. \( \frac{0 + a}{2} = 1 \) which gives \( a = 2 \)
2. \( \frac{0 + b}{2} = -2 \) which gives \( b = -4 \)

Thus, the coordinates of the other end of the diameter are \( (2, -4) \).

### Step 5: Finding the Equation of the Line
Now we need to find the equation of the line passing through the points \( (0, 0) \) and \( (2, -4) \). We can use the two-point form of the equation of a line:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]
Substituting \( (x_1, y_1) = (0, 0) \) and \( (x_2, y_2) = (2, -4) \):
\[ y - 0 = \frac{-4 - 0}{2 - 0}(x - 0) \]
This simplifies to:
\[ y = -2x \]

### Step 6: Rearranging to Standard Form
Rearranging \( y = -2x \) gives:
\[ 2x + y = 0 \]

### Final Answer
Thus, the equation of the diameter of the circle that passes through the origin is:
\[ \boxed{2x + y = 0} \]