The parametric equations of the circle `x^(2)+y^(2)+2x+4y-11=0` are

Options

A. `x=1+4cos theta,y=2+4sin theta`
B. `x=-1+4cos theta ,y=2+4sin theta`
C. `x=-1+4cos theta , y=-2+ 4 sin theta`
D. `x=1-4cos theta , y=2-4sin theta`

Answer

To find the parametric equations of the circle given by the equation \(x^2 + y^2 + 2x + 4y - 11 = 0\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
We start with the given equation of the circle:
\[ x^2 + y^2 + 2x + 4y - 11 = 0 \]
We can rearrange it to group the \(x\) and \(y\) terms:
\[ x^2 + 2x + y^2 + 4y = 11 \]

### Step 2: Complete the Square
Next, we complete the square for the \(x\) and \(y\) terms.

**For \(x\):**
\[ x^2 + 2x \quad \text{can be rewritten as} \quad (x + 1)^2 - 1 \]

**For \(y\):**
\[ y^2 + 4y \quad \text{can be rewritten as} \quad (y + 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ ((x + 1)^2 - 1) + ((y + 2)^2 - 4) = 11 \]
Simplifying this:
\[ (x + 1)^2 + (y + 2)^2 - 5 = 11 \]
\[ (x + 1)^2 + (y + 2)^2 = 16 \]

### Step 3: Identify the Center and Radius
From the equation \((x + 1)^2 + (y + 2)^2 = 16\), we can identify:
- The center of the circle \((h, k)\) is \((-1, -2)\).
- The radius \(r\) is \(\sqrt{16} = 4\).

### Step 4: Write the Parametric Equations
The parametric equations of a circle are given by:
\[ x = h + r \cos \theta \]
\[ y = k + r \sin \theta \]
Substituting \(h = -1\), \(k = -2\), and \(r = 4\):
\[ x = -1 + 4 \cos \theta \]
\[ y = -2 + 4 \sin \theta \]

### Final Parametric Equations
Thus, the parametric equations of the circle are:
\[ x = -1 + 4 \cos \theta \]
\[ y = -2 + 4 \sin \theta \]