Find the parametric representation of the circles :
(i) `3x^(2) + 3y^(2) = 4 `.
`(ii) ( x - 2)^(2) + (y - 3)^(2) = 5 `.

Answer

To find the parametric representation of the given circles, we will follow a systematic approach for each equation.

### (i) For the circle given by the equation \(3x^2 + 3y^2 = 4\):

1. **Rewrite the equation in standard form**:
\[ x^2 + y^2 = \frac{4}{3} \]
This is the standard form of a circle centered at \((0, 0)\) with radius \(r = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}\).

2. **Identify the center and radius**:
- Center \((h, k) = (0, 0)\)
- Radius \(r = \frac{2}{\sqrt{3}}\)

3. **Use the parametric equations for a circle**:
The parametric representation of a circle is given by:
\[ x = h + r \cos \theta \]
\[ y = k + r \sin \theta \]
Substituting the values of \(h\), \(k\), and \(r\):
\[ x = 0 + \frac{2}{\sqrt{3}} \cos \theta = \frac{2}{\sqrt{3}} \cos \theta \]
\[ y = 0 + \frac{2}{\sqrt{3}} \sin \theta = \frac{2}{\sqrt{3}} \sin \theta \]

4. **Final parametric representation**:
Therefore, the parametric representation of the first circle is:
\[ x = \frac{2}{\sqrt{3}} \cos \theta, \quad y = \frac{2}{\sqrt{3}} \sin \theta \]

### (ii) For the circle given by the equation \((x - 2)^2 + (y - 3)^2 = 5\):

1. **Identify the center and radius**:
The equation is already in standard form:
- Center \((h, k) = (2, 3)\)
- Radius \(r = \sqrt{5}\)

2. **Use the parametric equations for a circle**:
Using the same parametric representation:
\[ x = h + r \cos \theta \]
\[ y = k + r \sin \theta \]
Substituting the values of \(h\), \(k\), and \(r\):
\[ x = 2 + \sqrt{5} \cos \theta \]
\[ y = 3 + \sqrt{5} \sin \theta \]

3. **Final parametric representation**:
Therefore, the parametric representation of the second circle is:
\[ x = 2 + \sqrt{5} \cos \theta, \quad y = 3 + \sqrt{5} \sin \theta \]

### Summary of Parametric Representations:
1. For the first circle:
\[ x = \frac{2}{\sqrt{3}} \cos \theta, \quad y = \frac{2}{\sqrt{3}} \sin \theta \]
2. For the second circle:
\[ x = 2 + \sqrt{5} \cos \theta, \quad y = 3 + \sqrt{5} \sin \theta \]