Question

For the parabola \( y^2 = 16x \), find the distance between the focus and the directrix.

• A. \( 2 \)
• B. \( 4 \)
• C. \( 8 \)
• D. \( 16 \)

Hint:

For the parabola \( y^2=4ax \), remember the shortcut: \[ \text{Distance between focus and directrix}=2a \] This avoids unnecessary coordinate calculations.

Answer 1

The Correct Option is C

Concept: The standard equation of a parabola opening rightward is: \[ y^2=4ax \] For this parabola:
• Vertex \(=(0,0)\)
• Focus \(=(a,0)\)
• Directrix \(x=-a\) The distance between focus and directrix is always: \[ 2a \]

Step 1: Comparing with the standard equation.
Given: \[ y^2=16x \] Compare with: \[ y^2=4ax \] Thus: \[ 4a=16 \] Hence: \[ a=4 \]

Step 2: Finding the focus and directrix.
Focus: \[ (a,0)=(4,0) \] Directrix: \[ x=-4 \]

Step 3: Calculating the distance.
The perpendicular distance from point \((4,0)\) to line \(x=-4\) is: \[ |4-(-4)| \] Thus: \[ |8|=8 \] Therefore, \[ \boxed{8} \] Alternatively, directly using: \[ \text{Distance}=2a=2(4)=8 \]