Reflective Property of Parabolas

When a plane intersects a double-napped cone in a direction parallel to one of its sides, it forms a U-shaped curve called a parabola.

A parabola’s geometry makes each point equidistant from the focus and directrix; this structure causes incoming parallel rays to reflect toward the focus.

This principle is used in satellite dishes to direct incoming signals toward a receiver.

For instance, in a satellite dish that is 6 meters wide and 1 meter deep, the receiver is placed at the focus of the dish’s cross-section, located p units above the vertex.

To determine the receiver’s distance, the two-dimensional cross-section of the three-dimensional dish is modeled as a parabola that opens upward, with its vertex at the origin.

First, the point (3, 1) on the curve is identified—it represents half the dish's width and full depth.

Using the standard form x² = 4py and the point’s coordinates, the value of p is calculated.

Solving the equation gives p = 2.25 m, indicating that the focus—and therefore the receiver—is located 2.25 meters above the vertex.

This setup ensures an accurate concentration of signals at the receiver.