1. What is the Focal Point of a Parabola?

The focal point (or focus) of a parabola is a fixed point that, along with the directrix, defines the parabola. All points on the parabola are equidistant to the focus and the directrix.

2. How Does the Calculator Work?

The calculator uses the focal point formula:

Where:

• \( a \) — Coefficient in the parabola equation \( y = a(x-h)^2 + k \)
• \( h \) — x-coordinate of the vertex
• \( k \) — y-coordinate of the vertex
• \( p \) — Distance from vertex to focus

Explanation: The focal point is always located a distance \( p \) from the vertex along the axis of symmetry.

3. Importance of Focal Point

Details: The focal point is crucial in optics (parabolic mirrors and antennas), projectile motion analysis, and architectural design of parabolic structures.

4. Using the Calculator

Tips: Enter the coefficient 'a' from the parabola's equation (must not be zero), and the (h,k) vertex coordinates. The calculator will determine the focal point.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola opens horizontally?
A: For horizontal parabolas \( x = a(y-k)^2 + h \), the focal point is at \( (h + p, k) \).

Q2: What does the 'a' value represent?
A: The 'a' value determines the parabola's width and direction. Larger |a| means a narrower parabola; positive a opens upward, negative opens downward.

Q3: How is the focal length related to 'p'?
A: The focal length is equal to |p|, the distance between vertex and focus.

Q4: What's the relationship between focus and directrix?
A: The directrix is the line \( y = k - p \), equidistant from the vertex as the focus but in the opposite direction.

Q5: Can this calculator work for 3D paraboloids?
A: No, this calculates for 2D parabolas only. 3D paraboloids have more complex focal properties.