1. What Is The Focus Of A Parabola?

The focus of a parabola is a fixed point that, along with the directrix, defines the curve. All points on a parabola are equidistant from both the focus and the directrix. For a parabola in vertex form y = a(x-h)² + k, the focus is located at (h, k + 1/(4a)).

2. How Does The Calculator Work?

The calculator uses the focus formula:

Where:

• \( h \) — x-coordinate of the vertex
• \( k \) — y-coordinate of the vertex
• \( a \) — Coefficient in the parabola equation y = a(x-h)² + k

Explanation: The focus lies along the axis of symmetry of the parabola, at a distance of 1/(4a) from the vertex.

3. Importance Of Focus Calculation

Details: Determining the focus is essential in optics (parabolic reflectors), satellite dish design, and understanding the geometric properties of conic sections.

4. Using The Calculator

Tips: Enter the vertex coordinates (h, k) and the coefficient a from the parabola equation y = a(x-h)² + k. The coefficient a cannot be zero.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola opens horizontally instead of vertically?
A: For horizontally opening parabolas (x = a(y-k)² + h), the focus would be at (h + 1/(4a), k).

Q2: What happens when a = 0?
A: When a = 0, the equation becomes y = k, which is a horizontal line, not a parabola. The focus is undefined in this case.

Q3: How is the focus related to the directrix?
A: The directrix is a line located at y = k - 1/(4a) for vertically opening parabolas. The focus and directrix are equidistant from the vertex.

Q4: Can the focus be inside the parabola?
A: Yes, the focus is always located inside the curve of the parabola.

Q5: What are practical applications of parabolic focus?
A: Parabolic reflectors use the focusing property to concentrate light, sound, or radio waves at the focus point, used in telescopes, satellite dishes, and headlights.