Reflective Property Reflective Property of Parabolasof Parabolas

One more day in Sec. 8.1b…One more day in Sec. 8.1b…

But first, let’s start with a problem:

Prove that the graph of the given equation is a parabola, andfind its vertex, focus, and directrix.

V: (1, 7/6), F: (1, 5/3), D: V: (1, 7/6), F: (1, 5/3), D: yy = 2/3 = 2/3

23 6 6 10 0x x y

23 2 6 10x x y

23 2 1 6 10 3x x y

23 1 6 7x y

2 73 1 6

6x y

2 71 2

6x y

Write in standard form:

More Practice

Prove that the graph of the given equation is a parabola, andfind its vertex, focus, and directrix.

V: (13/4, 1), F: (9/4, 1), D: V: (13/4, 1), F: (9/4, 1), D: xx = 17/4 = 17/4

2 2 4 12 0y y x

2 2 4 12y y x 2 2 1 4 12 1y y x

21 4 13y x 2 13

1 44

y x

Write in standard form:

More PracticeWrite an equation for the parabola graphed below.

24y k p x h

V(1,–3)

(11/2,0)

General equation:

23 4 1y p x

Substitute in vertex:

20 3 4 5.5 1p

Substitute in point, solve for p:

4 2p 2

3 2 1y x Final Answer:

Reflective Property of Parabolas

If a parabola is rotated in 3-D space around its axis, theparabola “sweeps out” a paraboloid of revolution.

These are important because a signal (light, sound, radiowaves, etc.) emanating from the focus reflects off thesurface of the dish in lines parallel to the axis (and viceversa – incoming parallel waves are sent directly towardthe focus…).

Check out Figure 8.9 on p.637!!!Check out Figure 8.9 on p.637!!!

During football games, a TV network uses a parabolic reflectorwith a microphone at the reflector’s focus to capture theconversations among players on the field. If the parabolicreflector is 3 ft across and 1 ft deep, where should themicrophone be placed?

A diagram of a cross-sectionof the reflector:

Practice ProblemsPractice Problems

F(0, p)

V(0, 0)

(1.5, 1)(–1.5, 1)

2 4x pyGeneral equation:

Substitute the points:

21.5 4 1p

2.250.5625

4p

During football games, a TV network uses a parabolic reflectorwith a microphone at the reflector’s focus to capture theconversations among players on the field. If the parabolicreflector is 3 ft across and 1 ft deep, where should themicrophone be placed?

A diagram of a cross-sectionof the reflector:

Practice ProblemsPractice Problems

F(0, p)

V(0, 0)

(1.5, 1)(–1.5, 1)

Because p = 0.5625 ft,Because p = 0.5625 ft,or 6.75 inches, theor 6.75 inches, themicrophone should bemicrophone should beplaced inside theplaced inside thereflector along its axisreflector along its axisand 6.75 inches fromand 6.75 inches fromits vertex.its vertex.

The mirror of a flashlight is a paraboloid of revolution. Itsdiameter is 6 cm and its depth is 2 cm. How far from the vertexshould the filament of the light bulb be placed for the flashlightto have its beam run parallel to the axis of its mirror?

Practice ProblemsPractice Problems

Because p = 1.125 cm, the filament should be placedBecause p = 1.125 cm, the filament should be placed1.125 cm from the vertex along the axis of the mirror.1.125 cm from the vertex along the axis of the mirror.

A main cables of a suspension bridge uniformly distribute theweight of the bridge when in the form of a parabola. The maincables of a particular bridge are attached to towers that are600 ft apart. The cables are attached to the towers at a heightof 110 ft above the roadway and are 10 ft above the roadwayat their lowest points. If vertical support cables are at 50-ftintervals along the level roadway, what are the lengths of thesevertical cables?

Practice ProblemsPractice Problems

Lengths (in feet): 79.444, 54.444, 35, 21.111,Lengths (in feet): 79.444, 54.444, 35, 21.111,12.778, 10, 12.778, 21.111, 35, 54.444, 79.44412.778, 10, 12.778, 21.111, 35, 54.444, 79.444