Conics around us Some properties of conics (curves of the second degree)
Notes 8.2 Conics Sections – The Ellipse
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Transcript of Notes 8.2 Conics Sections – The Ellipse
Notes 8.2 Notes 8.2
Conics Sections –Conics Sections – The Ellipse The Ellipse
I. IntroductionI. Introduction
A.) A.) Def: The set of all points in a plane Def: The set of all points in a plane whose distances from two fixed points whose distances from two fixed points in the plane have a constant sum.in the plane have a constant sum.
1.) The fixed points are the 1.) The fixed points are the FOCIFOCI..2.) The line through the foci is the 2.) The line through the foci is the
FOCAL AXIS.FOCAL AXIS.3.) The 3.) The CENTER CENTER is ½ way is ½ way
between the foci and/or the vertices.between the foci and/or the vertices.
B.) B.) Forming an Ellipse - When a plane Forming an Ellipse - When a plane intersects a double-napped cone and is intersects a double-napped cone and is neither parallel nor perpendicular to the neither parallel nor perpendicular to the base of the cone, an ellipse is formed.base of the cone, an ellipse is formed.
P(x, y)
FocusFocus
d1 d2
(x, y)
(F1, 0) (F2, 0)
C.) C.) Pictures – By Definition - Pictures – By Definition -
Minor Axis
Focus
Focus
Major Axis
a is the SEMI-
MAJOR axis
Center
(-a, 0) (a, 0)
(0, -b)
(0, b)
(-c, 0) (c, 0)(0, 0)
b is the SEMI-
MINOR axis
Pictures -Expanded- Pictures -Expanded-
VertexVertex
2 2 2 2
2 2 2 21 or 1
x y y x
a b a b
Where b2 + c2 = a2.
D.) D.) Standard Form Equation - Standard Form Equation -
a
0, cy axisx axis
a
2 2
2 21
y x
a b
b
,0c
2 2
2 21
x y
a b St. Fm.
Focal axis
Foci
Semi-Major
Semi-Minor
Pyth. Rel. 2 2 2a b c
b2 2 2a b c
E.) ELLIPSES - E.) ELLIPSES - Center at (0,0) Center at (0,0)
a
,h k cx hy k
a
2 2
2 21
y k x h
a b
b
,h c k
2 2
2 21
x h y k
a b
St. fm.
Focal axis
Foci
Semi-Major
Semi-Minor
Pyth. Rel. 2 2 2a b c
b2 2 2a b c
F.) ELLIPSES - F.) ELLIPSES - Center at (Center at (hh, , kk) )
A. ) Ex. 1- Find the vertices and foci of the following ellipse:
II. ExamplesII. Examples
2c
27 5 c
2 2 2a b c
2 2
17 5
x y
2 25 7 35x y
Foci =
Vertices = 7,0 and 7,0
2,0 and 2,0
B.) Ex. 2- Find a equation of the ellipse with foci (4,0) and (-4,0) whose minor axis has a length of 6.
2 2
125 9
x y
5a
2 2 23 4a
4, 3c b
C.) Ex. 3- Find the center, foci, and vertices of the following ellipse:
foci : 3 7,5
center : 3,5
7c
216 9 c
2 23 5
116 9
x y
vertices : 7,5 & 1,5
D.) Ex. 2- Find the equation of an ellipse with foci (-2, 1) and (-2, 5) and major-axis endpoints (-2, -1) and (-2,7).
216 4b
4
2
a
c
center 2,3 foci 2,1 & 2,5
vertices. 2, 1 & 2,7
2 23 2
116 12
y x
III. EccentricityIII. Eccentricity
A.)A.)
B.) What it tells us – B.) What it tells us –
1.) 1.) ee close to 0 close to 0 foci close to center foci close to center
2.) e2.) e close to 1 close to 1 foci close to vertices foci close to vertices
0 1e c
ea
IV. Ellipsoids of RevolutionIV. Ellipsoids of Revolution
A.) Rotate ellipse about its focal axis to get an ellipsoid of revolution
B.) Examples of these include whispering galleries and a lithotripter, a device which uses shockwaves to destroy kidney stones.