Find the equation of the following curves in cartesian form. If the curve is a circle find the centres and radii. (i) `x = -1 + 2cosalpha, y = 3 + 2sinalpha.`

Answer

To find the equation of the curve given by the parametric equations \( x = -1 + 2 \cos \alpha \) and \( y = 3 + 2 \sin \alpha \), we will convert these equations into Cartesian form and identify if it represents a circle, along with its center and radius.

### Step-by-Step Solution:

1. **Write the given equations:**
\[ x = -1 + 2 \cos \alpha \]
\[ y = 3 + 2 \sin \alpha \]

2. **Rearrange the equations to express \(\cos \alpha\) and \(\sin \alpha\):**
- From the first equation:
\[ 2 \cos \alpha = x + 1 \implies \cos \alpha = \frac{x + 1}{2} \]
- From the second equation:
\[ 2 \sin \alpha = y - 3 \implies \sin \alpha = \frac{y - 3}{2} \]

3. **Use the Pythagorean identity:**
We know that:
\[ \cos^2 \alpha + \sin^2 \alpha = 1 \]
Substituting the expressions for \(\cos \alpha\) and \(\sin \alpha\):
\[ \left(\frac{x + 1}{2}\right)^2 + \left(\frac{y - 3}{2}\right)^2 = 1 \]

4. **Simplify the equation:**
\[ \frac{(x + 1)^2}{4} + \frac{(y - 3)^2}{4} = 1 \]
Multiply through by 4 to eliminate the denominators:
\[ (x + 1)^2 + (y - 3)^2 = 4 \]

5. **Identify the center and radius:**
The equation \((x + 1)^2 + (y - 3)^2 = 4\) is in the standard form of a circle:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

- Here, the center \((h, k)\) is \((-1, 3)\) and the radius \(r\) is given by:
\[ r = \sqrt{4} = 2 \]

### Final Answer:
The equation of the curve in Cartesian form is:
\[ (x + 1)^2 + (y - 3)^2 = 4 \]
- **Center:** \((-1, 3)\)
- **Radius:** \(2\)