Analytic Geometry

1
Analytic Geometry

• Dont Lose Your Focus!

2
Conic Sections
The intersections of a plane with a right
circular cone.
How we tilt the plane determines what conic
section appears.
3
Parabolae

• We know parabolae can be described using
  quadratics.
• What we dont know is that a parabola is actually
  the set of all points the same distance from a
  point (focus) as from a line (directrix).
• This means we can use the distance formula to
  determine an equation.

4
Parabola
Using this new definition, how can we write the
equation of a parabola? We simply equate the
distances and simplify.
The relevant equations are on page 670.
5
Parabola Features

• A parabola is defined by
• Vertex
• Focus (a,0) or (0,a)
• Directrix y a or x a
• The segment connecting the two points above and
  below the focus is called the latus rectum side
  straight
• Its length is 4a
• Graphing the latus rectum helps define the
  bowlnessof the parabola

6
Parabola Shifts

• Just like in previous chapters, rules for shifts
  apply.
• If the graph is shifted horizontally x-h
• If the graph is shifted vertically y-k
• Take your time figuring this out!
• First, determine symmetry
• Second, determine direction
• Third, determine shift

• To graph by hand
• Determine symmetry and how the parabola opens
• Determine the shift
• Determine a (the distance from vertex to focus)
• Plot the vertex, focus and latus rectum
• Plot the directrix
• Sketch the bowl
• Example p 678 29,55

7
Ellipses

• The common theme in conic sections (parabolae,
  ellipses, hyperbolae) is the focus or foci.
• For a parabola, it defines a point of
  equidistance.
• For an ellipse, two foci define a triangle
  equality.
• Sketchpad demo.

• Equation of an ellipse
• Essentially a modified pythagorean on a unit
  circle

8
Ellipses
If an ellipse lies along the x-axis, then the
major axis is x. If an ellipse lies along the
y-axis, the major axis is y. In the formula, a is
always the longer vertex, soIf an ellipse lies
along the y-axis, our equation becomes
Relevant equations are on pg. 687
9
Hyperbolas

• Again, we define a hyperbola with respect to its
  foci.
• The difference of two distances from one focus
  and another focus is constant twice the
  distance from center to vertex.
• A parabola sets two distances equal. An ellipse
  adds two distances and a hyperbola subtracts two
  distances.

10
Hyperbolas

• There are points in space for which no solution
  exists to the difference of distances
• The collections of these points are called the
  asymptotes of the hyperbola.

• As with ellipses, hyperbolas can lie along either
  axis. Since the tranverse axis is taken first,
  we simply switch x and y in our equation, which
  is very similar to ellipses

11
Hyperbolas
X is the transverse axis
No Shift Eq
Y is the transverse axis
Relevant equations are on pg. 702
12
Conics Together

• In all of our conic sections, we have squares and
  linears in some combination. This suggests a
  general form for conic sections, where we would
  set different things equal to zero, depending on
  which conic we were talking about.

13
General Form of a Conic
If B0, this equation is not too bad
Discuss Looking at the first two coefficients
only, can you figure out restrictions so the
general form reduces to a specific equation of an
ellipse, hyperbola or parabola? Hint Complete
the square if necessary.
Point Dont complete the square once weve
established this! Example pg717 2
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General Form of a Conic

• If AC0, we have a parabola
• If ACgt0, we have an addition, which implies an
  ellipse
• If AClt0, we have a subtraction, which implies a
  hyperbola
• To put in standard form, we would have to
  complete the square, as in previous sections.
  Dont. Think through the rules each time.

15
General Form with XY Term
This gets ugly because we cant complete the
square for x and y independently. That is, x and
y are coupled in the B term. Is there a way to
remove the B term, so the general form reduces to
what we know already? Yes rotate our
axes. Sketchpad
The goal is to line up the axes in a way that
is familiar. That is, remove the xy coupling of
the graph by rotating the axes so vertices and
foci lie on the axes, as in sections 2-4.
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General Form with XY Term
Before doing this rotation, we need to develop
some rules for rotation
To see where the cotangent restriction comes
from, well follow the derivation on page 713.
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General Form with XY Term
Well do Example 3, pg 714 together. Note For
all problems, do not solve for y and graph in the
TI! The algebra can be scary but remember the
XY-term will always cancel, if youve applied the
correct rotation (angle) to the coordinates.
As with Herons formula, we are applying an
algorithm to each problem. Because the algorithm
is the same, irrespective of the problem, there
ought to be a shortcut for determining what kind
of graph is what, without doing a rotation
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General Form with XY Term
the shortcut is to recognize an invariant term.
That is, a term that is unchanged by the rotation
A correct rotation will ensure B is zero, so
And the XY term disappears. We then analyze the
equation as in the first part of the section
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General Form with XY Term

• If then
• To summarize
• With no XY term, complete the square or use the
  shortcut
• With an XY term, apply a rotation or use the
  shortcut

20
Conics in Polar Form

• It seems reasonable that if conics are all loci
  involving foci, there should be a universal
  definition for the three conics discussed.
• This definition is a ratio of two distances, one
  involving a focus, the other a directrix.
• The ratio has a name eccentricity
• The eccentricity is also the ratio of focal
  length (c) to vertex length (a) in an ellipse and
  a hyperbola

21
Conics in Polar Form
What is the eccentricity of a parabola? For an
ellipse, a point on the ellipse will always be
closer to the focus than the directrix and
elt1 For a hyperbola, a point on the hyperbola
will always be closer to the directrix and egt1
22
Conics in Polar Form

• Problem 1 we havent defined a directrix for
  all conics.
• Solution 1 define it as a distance, p, from a
  vertex (just as a parabola)
• Problem 2 because eccentricity is now defined,
  how do we include it?
• Solution 2 define conics in polar form

23
Conics in Polar FormHavent we done this before?

• Conics in rectangular are generally defined
• Various terms equal zero, positive or negative to
  give us the 3 conics discussed in class.
• How do we translate to polar?
• Use conversions
• Start from scratch ?

• Use the pole as the reference (vertex)
• Define p as the distance from vertex to directrix
• Use polar expressions for distances
• Because two distances are proportional, not
  equal, we dont have to worry about separate
  cases for each conic.

24
d(D,P)
r
p
Pole O
(Focus F)
Derivation on the Board or Page 720 in text.
Directrix D
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Polar Conics Salient Features

• In terms of the radius
• Eccentricity shows up twice
• Use e from denominator to solve for p
• 1 must appear in denominator
• How would this change if
• Directrix is on opposite side of pole?
• Directrix is parallel to and above polar axis?
• Directrix is parallel to and below polar axis?
• Page 722 summarizes

26
Polar General Comparison
Extra Credit Get from polar to rectangular (or
vice versa) using
conversions from chapter 8.
27
Parametric Equations

• The motion along all of our conics can be modeled
  using separate functions for x and y.
• The functions we use contain a common variable,
  or parameter. Often, time
• An example is the motion of a projectile
• This motion is parabolic and described by a
  quadratic, either as in chapters 2-3 or chapter
  9.
• The parametric equation allows us to decouple x
  and y (look at each independent of the other)

28
Parametric Equations

• To graph on a TI, follow the blue box
  instructions on pg 726
• The finer your T-step, the better your curve
  looks
• Remember, x and y are now the result of a
  function involving t. x does not determine y!

• If you want to get back to rectangular form,
  simply solve one of the variable equations for t
  and back substitute into the other equation.
• Example Pg 736 12

29
Parametric Equations

• Common Applications
• Projectile motion
• Two simultaneous motions (two trains problem)
• Special curves
• Cycloid
• Inverted Cycloid
• Brachistochrone
• Tautochrone
• Motion on an elliptical curve. Pg 738 38

30
Parametric Equations

• Yet another way to represent curves
• Rectangular
• Polar
• Parametric
• Particularly useful for decoupling two or three
  dimensional motion problems
• Useful when rectangular coordinates are untenable