If (4,0) is the vertex , and Y-axis the directrix of a parabola , then its focus is

Options

A. (8,0)
B. (4,0)
C. (0,8)
D. (0,4)

Answer

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

### Step 1: Understand the given information
We know that the vertex of the parabola is at the point (4, 0) and the directrix is the Y-axis (which can be represented as the line x = 0).

### Step 2: Determine the orientation of the parabola
Since the directrix is vertical (the Y-axis), the parabola opens horizontally. Given that the vertex is at (4, 0), the parabola opens to the right.

### Step 3: Calculate the distance from the vertex to the directrix
The distance from the vertex (4, 0) to the directrix (the Y-axis, x = 0) is:
\[ \text{Distance} = 4 - 0 = 4 \]

### Step 4: Find the focus
The focus of a parabola is located at a distance equal to the distance from the vertex to the directrix, but in the opposite direction. Since the parabola opens to the right, we move 4 units to the right of the vertex.

Thus, the coordinates of the focus are:
\[ \text{Focus} = (4 + 4, 0) = (8, 0) \]

### Conclusion
The focus of the parabola is at the point (8, 0).

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