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, serves to define 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 for a parabola in vertex form:

Where:

• \( a \) — Coefficient that determines the parabola's width and direction
• \( h \) — x-coordinate of the vertex
• \( k \) — y-coordinate of the vertex

Explanation: The focal point is always located inside the parabola, at a distance of \( \frac{1}{4a} \) from the vertex along the axis of symmetry.

3. Importance of Focal Point

Details: The focal point is crucial in optics (parabolic reflectors), satellite dishes, and architectural designs. It's where parallel rays of light converge when reflected off the parabola.

4. Using the Calculator

Tips: Enter the coefficient 'a' (must be non-zero), and the vertex coordinates (h, k). The calculator will compute the focal point coordinates.

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 + 1/(4a), k).

Q2: How does the 'a' value affect the focal point?
A: Larger |a| values move the focus closer to the vertex, making the parabola "sharper."

Q3: What's the relationship between focus and directrix?
A: The directrix is the line y = k - 1/(4a), equidistant from the vertex as the focus but in the opposite direction.

Q4: Can a parabola have a=0?
A: No, a=0 would make it a linear equation, not a parabola.

Q5: How is this used in real-world applications?
A: Parabolic microphones, satellite dishes, and car headlights all use the focusing properties of parabolas.