Lesson 3: Focus of a Parabola

Property

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

Examples

• The path of a basketball shot through the air follows a parabolic arc due to gravity.
• A satellite dish is shaped like a parabola to gather incoming signals and reflect them to a single point, the focus, where the receiver is located.
• The reflector in a car's headlight has a parabolic shape to take the light from the bulb (at the focus) and project it forward in a strong, straight beam.

Explanation

Think of a parabola as a perfect U-shape. Every point on this curve is an equal distance away from a special point (the focus) and a special line (the directrix). This unique balance creates the parabola's signature curve.

Property

To derive a parabola equation, set the distance from any point $(x,y)$ to the focus equal to the distance from that point to the directrix. For a vertical parabola with focus at $(0,p)$ and directrix $y = -p$:

Property

For parabolas with vertex at the origin:

• Vertical parabolas:
  $$y = \frac{1}{4p}x^2$$
• Horizontal parabolas:
  $$x = \frac{1}{4p}y^2$$

where $p$ is the directed distance from vertex to focus.