1. What Is The Focus Of A Parabola?

The focus of a parabola is a fixed point that, along with the directrix, defines the parabolic shape. All points on the parabola are equidistant from the focus and the directrix.

2. How To Calculate The Focus?

For a parabola in vertex form:

The focus is calculated using:

Where:

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

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: Calculating the focus is essential in optics (parabolic reflectors), astronomy (telescope design), physics (projectile motion), and engineering (antenna design).

4. Using The Calculator

Tips: Enter the vertex coordinates (h, k) and the coefficient a. The coefficient a must not be zero. The calculator will compute the focus coordinates.

5. Frequently Asked Questions (FAQ)

Q1: What if the parabola opens horizontally instead of vertically?
A: For horizontally opening parabolas (x = a(y-k)² + h), the focus formula becomes (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 does the value of a affect the focus position?
A: Larger |a| values move the focus closer to the vertex, while smaller |a| values move it farther away. Negative a values indicate the parabola opens downward.

Q4: What is the relationship between focus and directrix?
A: The directrix is a line located at y = k - 1/(4a), exactly the same distance from the vertex as the focus but in the opposite direction.

Q5: Can this calculator handle parabolas that open left or right?
A: This calculator is designed for vertical parabolas (y = a(x-h)² + k). For horizontal parabolas, a different formula is needed.