introduction to conic sections. a conic section is a curve formed by the intersection of...
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Introduction to Conic Sections
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A conic section is a curve formed by the intersection of
_________________________a plane and a double cone.
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CirclesThe set of all points that are the same distance from the center.
Standard Equation: 222 )()( rkyhx
With CENTER: (h, k)
& RADIUS: r (square root)
(h , k)
r
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Example 1
81)8()2( 22 yx
-h-k r²
Center:
Radius: r
),(
9
k (), h 82
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Example 2
1)1( 22 yx
Center ?
Radius ?
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• Tilt a glass of water and the surface of the liquid acquires an elliptical outline.
• Salami is often cut obliquely to obtain elliptical slices which are larger.
The Ellipse
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• The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve.
• In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.
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• On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.
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• Any cylinder sliced on an angle will reveal an ellipse in cross-section
• (as seen in the Tycho Brahe Planetarium in Copenhagen).
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• The ellipse has an important property that is used in the reflection of light and sound waves.
• Any light or signal that starts at one focus will be reflected to the other focus.
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• The principle is also used in the construction of "whispering galleries" such as in St. Paul's Cathedral in London.
• If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.
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EllipseBasically an ellipse is a squished circle
Standard Equation: 1)()( 2
2
2
2
b
ky
a
hx
Center: (h , k)
a: major radius (horizontal), length from center to edge of circle
b: minor radius (vertical), length from center to top/bottom of circle
* You must square root the denominator
(h , k)
ab
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Example 3
14
)5(
25
)4( 22
yx
a² b
This must equal 1
Center: (-4 , 5)
a: 5
b: 2
2
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ParabolaWe’ve talked about this before…
a U-shaped graph
)(4)( 2 kyphx
Standard Equations:
)(4)( 2 hxpky OR
This equation opens up or down This equation opens left or right
HOW DO YOU TELL…LOOK FOR THE SQUARED VARIABLE
Vertex: (h , k)
•If there is a negative in front of the squared variable, then it opens down or left.
•If there is NOT a negative, then it opens up or right.
vertex
vertex
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• One of nature's best known approximations to parabolas is the path taken by a body projected upward, as in the parabolic trajectory of a golf ball.
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• The easiest way to visualize the path of a projectile is to observe a waterspout.
• Each molecule of water follows the same path and, therefore, reveals a picture of the curve.
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• This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.
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• Parabolas exhibit unusual and useful reflective properties.
• If a light is placed at the focus of a parabolic mirror, the light will be reflected in rays parallel to its axis.
• In this way a straight beam of light is formed.
• It is for this reason that parabolic surfaces are used for headlamp reflectors.
• The bulb is placed at the focus for the high beam and in front of the focus for the low beam.
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• The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space:
• ...the beam comes toward the parabolic surface and is brought into focus at the focal point.
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Example 4
)5()2(12
1 2 yx
What is the vertex? How does it open?(-2 , 5) opens down
Example 52)2(1255 yx
What is the vertex? How does it open?(0 , 2) opens right
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• If a right circular cone is intersected by a plane perpendicular to its axis, part of a hyperbola is formed.
• Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade.
The Hyperbola
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HyperbolasWhat I look like…two parabolas, back to back.
Standard Equations:
OR1
)()( 2
2
2
2
b
ky
a
hx1
)()( 2
2
2
2
b
hx
a
ky
Have I seen this before? Sort of…only now we have a minus sign in the middle
This equation opens up and downThis equation opens left and right
Center: (h , k)(h , k)
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Example 6
14
)5(
25
)4( 22
yx
Center: (-4 , 5)
Opens: Left and right
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Name the conic section and its center or vertex.
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2522 yx
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122 yx
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2)2(12
11 yx
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2)2(8
13 xy
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4)2( 22 yx
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1925
22
yx
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116
)2(
4
)1( 22
yx
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49)1()2( 22 yx
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1)7()5( 22 yx
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xy 62
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14
)1( 22
xy
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15
)5(
17
)4( 22
xy