Circle on which the co-ordinates of any point are `{2+4cos theta, -1+4 sin theta}` where `theta` is parameter is `(x-2)^(2)+(y+1)^(2)=16`.

Answer

To solve the problem, we need to show that the parametric coordinates given by \( (2 + 4 \cos \theta, -1 + 4 \sin \theta) \) represent a circle and that this circle is described by the equation \( (x - 2)^2 + (y + 1)^2 = 16 \).

### Step-by-Step Solution:

1. **Identify the parametric equations**:
- Given the coordinates:
\[ x = 2 + 4 \cos \theta \]
\[ y = -1 + 4 \sin \theta \]

2. **Rearrange the equations**:
- Rearranging for \( \cos \theta \) and \( \sin \theta \):
\[ x - 2 = 4 \cos \theta \quad \text{(1)} \]
\[ y + 1 = 4 \sin \theta \quad \text{(2)} \]

3. **Square both equations**:
- Squaring both sides of equations (1) and (2):
\[ (x - 2)^2 = (4 \cos \theta)^2 = 16 \cos^2 \theta \quad \text{(3)} \]
\[ (y + 1)^2 = (4 \sin \theta)^2 = 16 \sin^2 \theta \quad \text{(4)} \]

4. **Add the squared equations**:
- Adding equations (3) and (4):
\[ (x - 2)^2 + (y + 1)^2 = 16 \cos^2 \theta + 16 \sin^2 \theta \]

5. **Use the Pythagorean identity**:
- We know from trigonometry that \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ 16 \cos^2 \theta + 16 \sin^2 \theta = 16(1) = 16 \]

6. **Final equation**:
- Therefore, we have:
\[ (x - 2)^2 + (y + 1)^2 = 16 \]

7. **Conclusion**:
- This is the equation of a circle with center at \( (2, -1) \) and radius \( 4 \) (since \( \sqrt{16} = 4 \)). Thus, the parametric coordinates indeed represent the same circle described by the equation \( (x - 2)^2 + (y + 1)^2 = 16 \).