1. What is Focal Width of a Parabola?

The focal width (or latus rectum) of a parabola is the length of the line segment perpendicular to the axis of symmetry that passes through the focus and whose endpoints lie on the parabola.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( a \) — Coefficient in the parabola equation \( y = ax^2 \)

Explanation: For the standard parabola \( y = ax^2 \), the focal width is inversely proportional to the absolute value of the coefficient \( a \).

3. Importance of Focal Width

Details: The focal width helps determine key properties of a parabola including its focus, directrix, and overall shape. It's important in optics, antenna design, and physics applications.

4. Using the Calculator

Tips: Enter the coefficient \( a \) from the parabola equation \( y = ax^2 \). The value cannot be zero (as that would make it a straight line, not a parabola).

5. Frequently Asked Questions (FAQ)

Q1: How does focal width relate to focus?
A: The focal width is four times the distance from vertex to focus for standard parabolas.

Q2: What if my parabola isn't in standard form?
A: First convert to standard form \( y = ax^2 + bx + c \) by completing the square, then use the \( a \) coefficient.

Q3: Does focal width change with parabola orientation?
A: The formula changes slightly for horizontal parabolas (\( x = ay^2 \)), where focal width is still \( 1/|a| \).

Q4: What are practical applications of focal width?
A: Used in designing satellite dishes, headlights, and other reflective devices where precise focus is needed.

Q5: How does focal width affect parabola shape?
A: Smaller focal width means a "sharper" parabola, while larger focal width makes it "flatter".