If the parabola `y^(2) = 4ax` passes through the point (4, 1), then the distance of its focus the vertex of the parabola is

Options

A. `(1)/(16)`
B. `(1)/(4)`
C. 16
D. 4

Answer

To solve the problem, we need to find the distance between the focus and the vertex of the parabola given by the equation \(y^2 = 4ax\) and that passes through the point (4, 1).

### Step-by-Step Solution:

1. **Understand the Parabola Equation:**
The equation of the parabola is given as \(y^2 = 4ax\). In this equation, \(a\) represents the distance from the vertex to the focus.

2. **Identify the Vertex and Focus:**
The vertex of the parabola is at the origin (0, 0), and the focus is located at the point (a, 0).

3. **Substitute the Given Point:**
Since the parabola passes through the point (4, 1), we can substitute \(x = 4\) and \(y = 1\) into the parabola's equation:
\[ 1^2 = 4a(4) \]
Simplifying this gives:
\[ 1 = 16a \]

4. **Solve for \(a\):**
Rearranging the equation to solve for \(a\):
\[ a = \frac{1}{16} \]

5. **Calculate the Distance from Vertex to Focus:**
The distance from the vertex (0, 0) to the focus (a, 0) is simply the value of \(a\):
\[ \text{Distance} = a = \frac{1}{16} \]

### Final Answer:
The distance of the focus from the vertex of the parabola is \(\frac{1}{16}\).

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