Step 1: Understanding the Concept:
A parabola is defined geometrically as the set of all points that are equidistant from a fixed point called the focus and a fixed straight line called the directrix.
For a standard parabola of the form \( y^2 = 4ax \), the vertex is at the origin \( (0, 0) \).
The focus is located on the axis of symmetry (in this case, the x-axis) at a distance of '\( a \)' units from the vertex in the direction the parabola opens.
The directrix is a line perpendicular to the axis of symmetry, located at the same distance '\( a \)' from the vertex but in the opposite direction.
Consequently, the distance between the focus and the directrix is exactly twice the distance from the vertex to either one.
Step 2: Key Formula or Approach:
1. Compare the given equation \( y^2 = 16x \) with the standard form \( y^2 = 4ax \).
2. Identify the value of the parameter \( a \).
3. Use the focus-directrix distance formula: \( d = 2a \).
Step 3: Detailed Explanation:
The given equation of the parabola is:
\[ y^2 = 16x \]
Comparing this with the standard form:
\[ y^2 = 4ax \]
We equate the coefficients of \( x \):
\[ 4a = 16 \]
Dividing both sides by 4:
\[ a = 4 \]
Based on the properties of a standard parabola \( y^2 = 4ax \):
- The coordinates of the focus \( S \) are \( (a, 0) = (4, 0) \).
- The equation of the directrix line is \( x = -a \implies x = -4 \).
We want to find the distance between the point \( (4, 0) \) and the line \( x = -4 \).
Since the focus is a point and the directrix is a vertical line, the distance is simply the absolute difference between the x-coordinates:
\[ \text{Distance} = |x_{\text{focus}} - x_{\text{directrix}}| \]
\[ \text{Distance} = |4 - (-4)| \]
\[ \text{Distance} = |4 + 4| = 8 \]
Alternatively, we can use the direct shortcut:
\[ \text{Distance between focus and directrix} = 2a \]
Substituting \( a = 4 \):
\[ \text{Distance} = 2 \times 4 = 8 \text{ units} \]
This result matches Option (C).
Step 4: Final Answer:
By identifying that \( a = 4 \) for the parabola \( y^2 = 16x \), we applied the formula for the focus-directrix distance (\( 2a \)) to find the result is 8. This corresponds to Option (C).