Notes #3-___
Date:______
8.1Conic Sections and Parabolas (603)
I.Conic Sections
II.Parabola:the set of points in a plane equidistant from a
particular line (the directrix)and a particular
point (the focus) in the plane.
III. Equations:,a is the larger #
Ex.1
Center:
Vertices:
Foci:
Length of
Major Axis:
Minor Axis:
Eccentricity:
Now graph on your calculator. What equation(s) did you use?
What considerations need to occur?
Ex.2
Center:
Vertices:
Foci:
Length of
Major Axis:
Minor Axis:
Eccentricity:
Ex.3Write the standard eq: 9x2 + 5y2 + 36x – 30y + 36 = 0
Ex.4Write an equation if the foci are (0, -3) & (0, 3) and
the major axis has a length of .
Ex.5Write an equation if a focus is (0, -1) and a covertex is
(3, 3).
IV.Reflective Propertyof an Ellipse (651)
Notes #3-___
Date:______
All of this is reversed
if the focal axis is vertical. In an ellipse
“a” is associated with
the major axis.
a is the distance from the center to the vertex.
8.2Ellipses (614)
I.Ellipse:the locus of points where the sum of the
distances to two fixed points (foci) is constant.
d1 + d2 = d3+ d4
Use a piece of yarn to represent a fixed sum.
O:center (0, 0)Major axis: 2a Minor axis: 2b
V:vertices (-a, 0) & (a, 0) Length of semimajor axis: a
F:foci (-c, 0) & (c, 0)Length of semiminor axis: b
C:covertices (-b, 0) & (b, 0)
c2 = a2 – b2
Don’t confuse this e with the number e used with exponential and logarithmic eqs.
II.Eccentricity (roundness): What happens as c 0?
(Circle) 0 < e < 1 (Line segment)
III. Equations:,a is the larger #
Ex.1
Center:
Vertices:
Foci:
Length of
Major Axis:
Minor Axis:
Eccentricity:
Now graph on your calculator. What equation(s) did you use?
What considerations need to occur?
Ex.2
Center:
Vertices:
Foci:
Length of
Major Axis:
Minor Axis:
Eccentricity:
General form.
Always sketch the given information.
Summary:
Ex.3Write the standard eq: 9x2 + 5y2 + 36x – 30y + 36 = 0
Ex.4Write an equation if the foci are (0, -3) & (0, 3) and
the major axis has a length of .
Ex.5Write an equation if a focus is (0, -1) and a covertex is
(3, 3).
IV.Reflective Property of an Ellipse (651)
There is a whisper gallery at the VLA in New Mexico.
Notes #3-___
Date:______
a is the positive term, it has nothing to do with which one is larger. a is the distance from the center to the vertex.
Hyperbola: Foci are outside (farther) +
Ellipse: Foci are inside (closer) –
8.3Hyperbolas (624)
A hyperbola is the set of all points in a plane whose distances from two fixed points (foci) in the plane have a constant difference (2a).
Standard form:
Distance to foci:
c2 = a2 + b2
Asymptotes:
y – k = (x – h)
Transverse axis: 2asemitransverse axis: a
Conjugate axis: 2bsemiconjugate axis: b
If y is + it opens up/down:asymptotes:
y – k = (x – h)
Why is e > 1?
Complete the square to transform general form to standard.
Eccentricity: What happens as c approaches 1?
Ex.1
Center:
Vertices:
Foci:
Asymptotes:
Eccentricity:
Ex.2
Center:
Vertices:
Foci:
Asymptotes:
Eccentricity:
Ex.3Write the standard eq: x2 – 4y2 – 2x – 16y – 11 = 0.
Center:
Vertices:
Foci:
Asymptotes:
Eccentricity:
Always sketch the given information.
Ex.4Write an equation if the foci are (4, 0) & (4, 10) and
the vertices are (4, 1) & (4, 9).
Ex.5Write an equation if the foci are (0, ) and
the equation of an asymptote is 2y = 3x.
Ex.6Write an equation if the vertices are (-2, 1) & (2, 1)
and it goes through (5, 4).
Ex.7Write with parametric equations:
x = y = 5 + 2tant