Find the parametic equation of the circle : `x^(2)+y^(2)+px+qy=0`.

Answer

To find the parametric equations of the circle given by the equation \(x^2 + y^2 + px + qy = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
We start with the equation:
\[ x^2 + y^2 + px + qy = 0 \]
We can rearrange this equation to prepare for completing the square.

### Step 2: Complete the Square
We will complete the square for both \(x\) and \(y\):

1. For \(x\):
\[ x^2 + px = \left(x + \frac{p}{2}\right)^2 - \frac{p^2}{4} \]

2. For \(y\):
\[ y^2 + qy = \left(y + \frac{q}{2}\right)^2 - \frac{q^2}{4} \]

Now substituting these back into the original equation gives:
\[ \left(x + \frac{p}{2}\right)^2 - \frac{p^2}{4} + \left(y + \frac{q}{2}\right)^2 - \frac{q^2}{4} = 0 \]

### Step 3: Simplify the Equation
Rearranging the equation, we have:
\[ \left(x + \frac{p}{2}\right)^2 + \left(y + \frac{q}{2}\right)^2 = \frac{p^2 + q^2}{4} \]

### Step 4: Identify the Center and Radius
From the equation \(\left(x + \frac{p}{2}\right)^2 + \left(y + \frac{q}{2}\right)^2 = r^2\), we can identify:
- Center \((x_1, y_1) = \left(-\frac{p}{2}, -\frac{q}{2}\right)\)
- Radius \(r = \sqrt{\frac{p^2 + q^2}{4}} = \frac{\sqrt{p^2 + q^2}}{2}\)

### Step 5: Write the Parametric Equations
The parametric equations of the circle can be expressed as:
\[ x = x_1 + r \cos \theta \]
\[ y = y_1 + r \sin \theta \]

Substituting the values of \(x_1\), \(y_1\), and \(r\):
\[ x = -\frac{p}{2} + \frac{\sqrt{p^2 + q^2}}{2} \cos \theta \]
\[ y = -\frac{q}{2} + \frac{\sqrt{p^2 + q^2}}{2} \sin \theta \]

### Final Parametric Equations
Thus, the parametric equations of the circle are:
\[ x = -\frac{p}{2} + \frac{\sqrt{p^2 + q^2}}{2} \cos \theta \]
\[ y = -\frac{q}{2} + \frac{\sqrt{p^2 + q^2}}{2} \sin \theta \]