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 curve. 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 standard parabola equations:

Where:

• \( a \) — Coefficient determining 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' (cannot be zero), and the vertex coordinates (h, k). The calculator will determine the exact focal point of the parabola.

5. Frequently Asked Questions (FAQ)

Q1: What if my parabola opens horizontally?
A: For \( x = a(y - k)^2 + h \), the focal point is \( (h + \frac{1}{4a}, k) \).

Q2: What happens when 'a' is negative?
A: The parabola opens downward, and the focal point will be below the vertex.

Q3: How does 'a' affect the focal point?
A: Larger |a| values mean the focal point is closer to the vertex, creating a "sharper" parabola.

Q4: What's the relationship between focus and directrix?
A: The directrix is the line \( y = k - \frac{1}{4a} \), equidistant from the vertex as the focus.

Q5: Can this be used for 3D paraboloids?
A: The same principle applies, but in three dimensions the focus becomes a focal line or point in space.