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The Parabola

Why study the parabola?

The parabola has many applications in situations where:

  • Radiation often needs to be concentrated at one point (e.g. radio telescopes, pay TV dishes, solar radiation collectors) or
  • Radiation needs to be transmitted from a single point into a wide parallel beam (e.g. headlight reflectors).

Here is an animation showing how parallel radio waves are collected by a parabolic antenna. The parallel rays reflect off the antenna and meet at a point (the red dot, labelled F), called the focus.

Definition of a Parabola

The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).The word locus means the set of points satisfying a given condition.

In the following graph,

  • The focus of the parabola is at (0,p).
  • The directrix is the line =−p.
  • The focal distance is ∣p∣ (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.)
  • The point (x, y) represents any point on the curve.
  • The distance d from any point (x, y) to the focus (0,p) is the same as the distance from (x, y) to the directrix.
  • The axis of symmetry of this parabola is the y-axis.

the formula for a parabola:

x2 = 4py

In more familiar form, with "y = " on the left, we can write this as:

y = x2/ 4p

where p is the focal distance of the parabola.

A hyperbola is a pair of symmetrical open curves. It is what we get when we slice a pair of vertical joined cones with a vertical plane.

The Hyperbola

How do we create a hyperbola?

Take 2 fixed points A and B and let them be 4a units apart. Now, take half of that distance (i.e. 2a units).

Now, move along a curve such that from any point on the curve,

(distance to A) − (distance to B) =2a units.

The curve that results is called a hyperbola. There are two parts to the curve.

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