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YOU WILL LEARN ABOUT:

How conic sections are defined and where they are used.

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When a plane intersects a cone, a conic section is formed. The angle at which the plane intersects the cone determines what kind of conic section results: a parabola, an ellipse, a circle, or a hyperbola. Conic sections occur in nature, and they are often used in engineering projects. Radio telescopes, for example, have parabolic shapes.

Conic sections occur throughout nature. They are also ideally suited for many real-world applications.

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Car headlights have parabolic cross sections. Because the bulb is located at the focus, all of the reflected light rays form a single beam.
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A lithotripter is a machine that uses high-energy shock waves to destroy kidney stones. The source of the shock waves is at one focus of an ellipse, and the patient is positioned so the kidney stone is at the other focus.
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Some navigational instruments, like those based on LORAN, use hyperbolas to locate ships at sea.

CONIC SECTIONS
Equation of the parabola is

directrix: y = -2

Vertex:
V ( 0.00 , 0.00 )


Focus:
F ( 0.0 , 2.0 )


Equation of the ellipse is

( x − 4.00 ) 2 25.00 + y 2 20.00 = 1

Vertices:

V 1 ( − 1 , 0.00 )
V 2 ( 9.00 , 0.00 )

Foci:

F 1 ( 0.00 , 0.00 )
F 2 ( 8.00 , 0.00 )
x 2 9.00- y2 16.00=1
y2 9.00- x2 16.00=1

Asymptotes: y = ±26x

Vertices:
V 1 ( -3 , -0.00 )
V 2 ( 3 , 0.00 )

Foci:
F 1 ( -5 , -0.00 )
F 2 ( 5 , 0.00 )

Equation of the line tangent to the graph of (x)  : y = (x − )

INSTRUCTIONS

On the graph, move the points V and F and the directrix to see how the parabola changes.

PRACTICE QUESTIONS

1 2 3
1. If a parabola has vertex ( 0 , 0 ) and focus ( 0 , - 3 ) , what is the equation of its directrix?
2. What is the equation of the parabola with vertex ( 2 , 1 ) and directrix x=-1?
3. What is the focus of the parabola ( x − 2 ) 2 = 8 ( y + 3 ) ?

INSTRUCTIONS

In this diagram of a planet's orbit around a star, the x- and y-axes are measured in astronomical units (au), where 1 au ≈150 million km. One focus of the orbit, F1(indicated by the star), is fixed at the origin. Move the points V1,V2, and F2 to see how the planet's elliptical orbit changes.

PRACTICE QUESTIONS

1 2 3
1. If the orbit of the planet is such that one focus is at the origin, the other focus is at ( 2 , 0 ) , and the orbit passes through ( - 1 , 0 ) , what is the farthest the planet can be from the star?
2. If the orbit of the planet is such that one focus is at the origin, the other focus is at ( 6.5 , 0 ) , and the orbit passes through ( 7 , 0 ) , what is the closest the planet can be to the star?
3. If the minimum distance between the planet and the star is 5 au and the maximum distance is 9.5 au, what equation describes the planet's orbit?

INSTRUCTIONS

On the graph, move the points V1, V2, F1, and F2 to see how the hyperbola changes. The hyperbola can also be changed by moving the sliders for a and c. (Click the play button to see what happens when one of a and c is held constant and the other increases.)

PRACTICE QUESTIONS

1 2 3
1. What happens to the graph of a hyperbola that opens left and right if a is held constant and b increases?
2. What happens to the graph of a hyperbola that opens up and down if c is held constant and a decreases?
3. If a hyperbola centered at the origin has one focus at 0,5 and one vertex at 0,-3, what are the equations of the hyperbola's asymptotes?
Result

x can be any number from to