The co-ordinates of any point on the circle `x^2+y^2=4` are

Options

A. `(cosalpha,sinalpha)`
B. `(4cosalpha,4sinalpha)`
C. `(2cosalpha,2sinalpha)`
D. `(sinalpha,cosalpha)`

Answer

To find the coordinates of any point on the circle defined by the equation \(x^2 + y^2 = 4\), we can follow these steps:

### Step 1: Identify the standard form of the circle's equation
The given equation of the circle is \(x^2 + y^2 = 4\). This can be rewritten in the standard form of a circle's equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. Here, we can see that:
- The center \((h, k) = (0, 0)\)
- The radius \(r = \sqrt{4} = 2\)

### Step 2: Write the parametric equations for the circle
Using the standard parametric equations for a circle, we have:
\[ x = h + r \cos \theta \]
\[ y = k + r \sin \theta \]
Substituting the values of \(h\), \(k\), and \(r\):
\[ x = 0 + 2 \cos \theta = 2 \cos \theta \]
\[ y = 0 + 2 \sin \theta = 2 \sin \theta \]

### Step 3: Express the coordinates of any point on the circle
Thus, the coordinates of any point on the circle can be expressed as:
\[ (x, y) = (2 \cos \theta, 2 \sin \theta) \]
where \(\theta\) is any angle.

### Final Answer
The coordinates of any point on the circle \(x^2 + y^2 = 4\) are given by:
\[ (2 \cos \theta, 2 \sin \theta) \]