If the vertex of a parabola is (0,2), directrix on x-axis then its focus is :

Options

A. (0,0)
B. (0,4)
C. (4,0)
D. None of these.

Answer

To find the focus of the parabola given its vertex and directrix, we can follow these steps:

### Step 1: Understand the given information
The vertex of the parabola is given as (0, 2), and the directrix is on the x-axis. This means the directrix is the line y = 0.

### Step 2: Identify the orientation of the parabola
Since the vertex is above the directrix, the parabola opens upwards. The standard form of a parabola that opens upwards is given by the equation:
\[ (y - k) = \frac{1}{4p}(x - h)^2 \]
where (h, k) is the vertex and p is the distance from the vertex to the focus.

### Step 3: Set the vertex coordinates
From the vertex (0, 2), we have:
- h = 0
- k = 2

### Step 4: Determine the directrix and focus
The directrix is y = 0, which means the distance from the vertex to the directrix is:
\[ \text{Distance} = 2 - 0 = 2 \]
This distance is equal to p. Since the parabola opens upwards, the focus will be located p units above the vertex.

### Step 5: Calculate the focus
Since p = 2, we can find the coordinates of the focus:
\[ \text{Focus} = (h, k + p) = (0, 2 + 2) = (0, 4) \]

### Conclusion
Thus, the coordinates of the focus of the parabola are (0, 4).

### Final Answer
The focus of the parabola is **(0, 4)**.

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