Parabola Worksheet / Name ______

Warm-up exercise –the Circle

Analytic geometry is used to prove geometric theorems using algebra. Another powerful application of analytic geometry is the derivation of formulas that describe geometric objects algebraically. For example, consider the circle. A circleis fully describedby two parameters: a point (its center) and a distance (its radius). Given the point and the distance , a circleconsists of all points that are units away from point .

Whenever we use analytic geometry, the first step is decide where to place the coordinate system; let’s agree to place the origin at the point . Next we need to express the circle’s definition algebraically: if point is on the circle then it is r units from the origin; using the Pythagorean Theorem, we have that . Done, in one step!!!

  1. Suppose that we had placed our coordinate system so that the center of the circle was located at instead of at the origin. Derive the equation for such a circle.

The Parabola

You haveprobablybeen told that the graph of a quadratic equation is called a parabola. You know that the shape of a parabola is sort of “cup-like”, that it is symmetric about a vertical line that passes through the vertex of the parabola, etc.

However, the parabola was so-named long before algebra was invented; here is a brief history:

  • Menaechmus (380 BCE–320 BCE)described the parabola as a cross-section of a cone
  • Apollonius (262 BCE–190 BCE) named the parabola
  • Pappus (290–350)described the parabola in terms of its focus and directrix
  • Galileo (1564–1642)showed that objects falling due to gravity follow parabolic paths
  • Newton (1643–1727) made the first telescope using a parabolic mirror

Pappus gave a description for the parabola that is similar in character to the definition of a circle given earlier. A parabola is fully described by two parameters: a point (its focus) and a line (its directrix). Given the point and the line , a parabola consists of all points that are equally distant from and .

  1. This exercise will help you to better understand the definition given above. Draw a point on the graph above and carefully measure the distance from your point to F, and also measure the distance from your point to the line d; do they agree? Repeat at least twice more.
  1. Distance to F: ______Distance to d: ______
  1. Distance to F: ______Distance to d: ______
  1. Distance to F: ______Distance to d: ______

To derive the formula for the parabola using analytic geometry, we must first decide where to place the coordinate system; this time I am going to choose to place the y-axis through and perpendicular to , and the x-axis parallel to the line and halfway between and .

The coordinates for the point are, and so the equation for the directrix is (recall that I placed the x-axis parallel to and halfway between and ). See the drawing below:

Next we need to express the parabola definition algebraically: if point is on the parabola then the distance from to is equal to the distance from to d.

All that is left is to simplify our expression and solve it for y:

You learned in an earlier class that the coefficient of determines how shallow or deep the parabola is; you now know that it contains information about the location of the focus and directrix of the parabola!

  1. For each of the parabola equations below, determine the coordinates of the focus and the equation of the directrix:
  2. : Focus: ______Directrix: ______
  3. : Focus: ______Directrix: ______
  4. : Focus: ______Directrix: ______

Constructing points on a parabola

Given the focus and directrix of a parabola, how might we go about constructing points on the parabola (using only our compass and straightedge)? Recall that, by definition, a point is on the parabola if it is equidistant from and .

First, construct the perpendicular to at some point on the directrix. Next, construct the perpendicular bisector of segment . Since the point is on the perpendicular bisector we have that , and so must be a point on the parabola (see the drawing below).

  1. Using your compass and straightedge, construct at least 2 points on the parabola defined by the focus and directrix given in the drawing on the right.

The Reflection Property of the parabola

Avery useful feature of the parabola is called its reflection property: for any parabola, rays that are parallel to the axis of symmetry will be reflected through the focus (see figure below).

Perhaps the hardest part of the proof of this property is in establishing how reflections work with curved mirrors. For flat mirrors, the Law of Reflection tells us that the angle of incidence is equal to the angle of reflection. For a curved mirror, a ray reflects as it would from a flat mirror that is tangent to the curve at the point of incidence.

The following drawing comes from the construction method described above. A ray is parallel to the axis of symmetry and strikes the parabola at . By the construction by , and so . Also, because they are vertical angles, and so we have that , the angle of incidence equals the angle of reflection.

The last detail is to observe that the line is in fact tangent to the parabola. If it were not, would intersect the parabola at another point . Since is on we have , but since is also on the parabola is equal to the distance from to d, which is impossible. Therefore must be a tangent to the parabola, and the proof is complete.

Parabola Trivia

Liquid mirror telescopes

The surface of a liquid in a rotating container will be parabolic. This fact has been used to make liquid mirror telescopes using mercury; compared to a solid glass mirror that must be cast, ground, and polished, a rotating liquid metal mirror is vastly cheaper to manufacture.

Parabolas in nature and machines

The path of any thrown object is a parabola, from baseballs to bullets to water from a hose. The arc of the cable of a suspension bridge is a parabola. The dish of a radio telescope, flashlight and automobile headlamp reflectors, telescope mirrors, satellite dishes and “big-ear” microphones used on the football field are all examples of applications of the parabola.

All parabolas are similar

Similar figures are shapes that are scale-models of each other; for example, all circles are similar figures. Likewise the network of roads and a city map differ only in size, and so are similar. It turns out that all parabolas are similar to each other; what appear to be different curves can be made to coincide by simply zooming in our out by an appropriate amount!