1. What is a Parabola's Focus and Directrix?

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). For the standard parabola x²=4ay, the focus is at (a,0) and the directrix is the line x=-a.

2. How Does the Calculator Work?

The calculator uses the standard parabola equation:

Where:

• \( a \) — Distance from vertex to focus (and vertex to directrix)
• Focus — Located at \( (a, 0) \)
• Directrix — The line \( x = -a \)

Explanation: The parabola opens to the right if a > 0 or to the left if a < 0. All points on the parabola are equidistant from the focus and directrix.

3. Importance of Focus and Directrix

Details: The focus-directrix property defines the parabola's shape. This is fundamental in optics (parabolic mirrors), antenna design, and projectile motion analysis.

4. Using the Calculator

Tips: Enter the value of 'a' from the standard equation x²=4ay. The calculator will determine the focus, directrix, and display the standard equation.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola equation is different?
A: This calculator works for standard form x²=4ay. Other forms require completing the square to convert to standard form.

Q2: What does a negative 'a' value mean?
A: A negative 'a' means the parabola opens to the left instead of the right.

Q3: How is this related to parabolic mirrors?
A: In parabolic mirrors, light rays parallel to the axis reflect through the focus, and vice versa.

Q4: Can this calculator handle vertical parabolas?
A: No, this is specifically for horizontal parabolas (x²=4ay). Vertical parabolas have the form y²=4ax.

Q5: What's the relationship between 'a' and the parabola's width?
A: Smaller |a| values make the parabola wider, larger |a| values make it narrower.