Question

The focus of the parabola \( y^2 = 16x \) is

• A. \( (4, 0) \)
• B. \( (0, 4) \)
• C. \( (2, 0) \)
• D. \( (0, 2) \)

Hint:

For standard parabolas of the form \( y^2 = 4ax \), the focus is always at \( (a, 0) \) and the directrix is the vertical line \( x = -a \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to determine the geometric coordinates of the focus for the given parabola equation \( y^2 = 16x \).


Step 2: Key Formula or Approach:
The standard equation of a rightward opening horizontal parabola is \( y^2 = 4ax \).
For this standard form, the focus is located at the coordinates \( (a, 0) \).


Step 3: Detailed Explanation:
Compare the given equation \( y^2 = 16x \) with the standard form \( y^2 = 4ax \).
Equating the coefficients of the \( x \) term:
\[ 4a = 16 \] \[ a = \frac{16}{4} = 4 \] Since the parabola is of the form \( y^2 = 4ax \), its focus is at \( (a, 0) \).
Substitute \( a = 4 \) to get the coordinates of the focus:
Focus = \( (4, 0) \).


Step 4: Final Answer:
The focus of the parabola is \( (4, 0) \).