Transcript 2001 ICTM Contest Division A Orals Topic: The Lore of

2003 ICTM Contest Division AA
Orals Topic: Conics
Micah Fogel
Illinois Math and Science Academy
The Source
There is no official source this year.
• Source was to be Schaum’s Outline:
Analytic Geometry
• Out of print!
Topics
•
•
•
•
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Locus definitions of conics
Focus/directrix definitions of conics
Rectangular equations
Parametric equations
Translation and rotation of axes
Topics (State level)
• Polar forms for conics
Additional Resources
• Mailing list: [email protected]
• Webstie: http://www.imsa.edu/~fogel
• ICTM Website:
www.viebach.net/ictm/admin/c2003r.htm
• Books
– Analytic geometry
– Calculus
– Last few orals sources
Study Ideas
• Study each aspect by itself (i.e., rectangular
equations one day, proofs of reflection
properties another day, etc.)
• Steal lots of basic exercises so that the
algebra becomes automatic
• Try to find different explanations of the
relationships between the various properties
of conics
Cones and Conic Sections
Slicing a Cone
Intersect a double-cone with a plane. If the
plane goes through the vertex of the cones,
the intersection is either a point, a line, or
two crossing lines.
Slicing a Cone
• These three cases are degenerate and are
not really very interesting
• The interesting stuff happens when the
plane does not go through the vertex
• Then there are three cases
– Plane has less slant than cone
– Plane has more slant than cone
– Plane is parallel to slant of cone
Ellipse
• In the first case, there
are two spheres, called
Dandelin (Belgian,
1794–1847) spheres,
which are tangent to
both cone and plane
• The points where
these spheres touch the
plane will be called
foci (sing.: focus)
Ellipse
• The intersection of the
plane with the cone is
called an ellipse
• Pick a point on the
ellipse and draw the
line through it and the
vertex of the cone
• The distance between
the two dashed circles
on this line is constant
Ellipse
• But the distances to
the foci are the same
as the distances along
this line, since they are
tangents to the same
spheres
• Thus, for any point on
an ellipse, the sum of
the distances to the
foci is a constant
Ellipse
• Concentrate on one of
the Dandelin spheres.
Consider the plane
through the circle
where it is tangent to
the cone
• Call the line where
this plane intersects
the cutting plane a
directrix
Ellipse
• From a point P on the
ellipse, travel parallel
to the axis of the cone
to the new plane, call
this point Q. Travel
along the cone until
the intersection of the
cone and sphere is
reached. Call this
point A. Choose D so
that QD  directrix
Q
A
P
D
F
Ellipse
• Note that in triangle
PQD, we have
PQ = PD cos , where
 is the angle between
the cutting plane and
the cone axis
D
Q

P
Q
D
P
Ellipse
• Also note that angle
QPA has the same
measure as the angle
between the axis and
slant of the cone. So
PQ = PA cos 
• Also note PF = PA,
since both are tangents
to a sphere from P
Q
A
P
D
F
Ellipse
• Putting these relationships together gives
PD cos  = PF cos , or that
PF cos 

PD cos 
• Thus, since  and  are constants, the ratio
of the distances from a point on the ellipse
to a focus and to the corresponding directrix
is a constant, called the eccentricity
Ellipse—Summary
• An ellipse is the intersection of a cone and a
plane with less slant than the cone
• The ellipse has two foci, which are the
intersection of the slant plane and the
Dandelin spheres
• The sum of the distances from the foci to
any point on the ellipse is a constant
Ellipse—Summary
• The slant plane meets the plane through the
circle where a Dandelin sphere intersects
the cone in a line called the directrix
• The distance from any point on the ellipse
to a focus is a constant—the eccentricity—
times the distance from that point to the
directrix
• Since the plane has less slant than the cone,
the eccentricity is less than one
Hyperbola
• If the slant of the plane is more than the
slant of the cone, then the plane intersects
both nappes of the cone
• This time, both Dandelin spheres are on the
same side of the plane
• Thus, everything works the same, except
that we have to subtract distances to foci to
get a constant, and the eccentricity will be
larger than one
Hyperbola—Summary
• A hyperbola is the intersection of a cone
and a plane with more slant than the cone
• The hyperbola has two foci, which are the
intersection of the slant plane and the
Dandelin spheres
• The difference of the distances from the foci
to any point on the hyperbola is a constant
Hyperbola—Summary
• The slant plane meets the plane through the
circle where a Dandelin sphere intersects
the cone in a line called the directrix
• The distance from any point on the
hyperbola to a focus is a constant—the
eccentricity— times the distance from that
point to the directrix
• Since the plane has more slant than the
cone, the eccentricity is greater than one
Parabola
• When the slant plane has the same slant as
the cone itself, the intersection is a single
curve called a parabola
• This time, there can only be one Dandelin
sphere
• So a parabola has only one focus
Parabola
• We can repeat all the same arguments and
calculations to find:
– A parabola is the intersection of a cone with a
plane of the same slant
– Because slants are equal, there is only one
Dandelin sphere, which meets the plane in the
focus
– Because slants are equal, eccentricity is one, so
parabola is the set of points equidistant from a
line (the directrix) and a point (the focus)
Circle
• A special case of a plane intersecting a cone
is if the plane is perpendicular to the axis of
the cone, resulting in a circle
• If we repeat all the above constructions,
there are two Dandelin spheres, which are
both tangent to the cutting plane—circles
have only one “focus,” the center
• The relevant planes are parallel, so circles
have no directrices. Eccentricity is zero
Reflection Principle
• Given two points on the same side of a line,
the shortest path from one to the other
which touches the line meets the line at
equal angles coming and going
• This can be seen by reflecting one point
across the line, and knowing the shortest
distance between the other point and the
reflection is a straight line
Reflection Principle
P2’
P2
P1
Reflection Propety of Ellipses
• Apply this to the foci and a tangent line to
an ellipse
Q
P’
F1
P
F2
• F1QF2 > F1PF2 since F1P’F2 = F1PF2, so P
is the point at which incident angles are
equal
Reflection Property of Parabolas
• If we move the focus F2 more and more to
the right, the reflected ray becomes more
and more horizontal
• If we let F2 “go to infinity” the ellipse turns
into a parabola
• So a ray from the focus of a parabola will
be reflected parallel to the axis of the
parabola
Reflection Property of
Hyperbolas
• If you keep moving F2 to the right, “past”
infinity, it reappears on the left. The ellipse
turns inside-out, and becomes a hyperbola
• So the corresponding reflection property for
hyperbolas is that a ray from one focus will
be reflected off the hyperbola directly away
from the other focus
(Of course, these aren’t rigorous proofs for
parabola and hyperbola!)
Reflection Property of
Hyperbolas
Cartesian Equations for Conics
• We can use the distance formula and the
focal distance properties of conics to find
equations for their graphs in the Cartesian
plane
• For instance, a circle with radius r and
2
2
center (h, k) is simply (x  h)  (y  k)  r
• Squaring, we obtain the standard equation
(x - h)2 + (y - k)2 = r2
Cartesian Equation for Ellipse
• If we put the foci at (c, 0) and make the
total distance of a point to the two foci 2a
the distance formula gives us
(x  c)  y  (x  c)  y  2a
• Move one radical to the other side, square,
and simplify to obtain
2
2
2
2
2cx  2cx  4a (x  c)  y  4a
2
2
2
Cartesian Equation for Ellipse
• Isolate the radical and square again:
a2(x2 + 2cx + c2 + y2) = a4 + 2a2cx + c2x2
2
• This simplifies to
(with b2 = a2 - c2)
• Eccentricity e = c/a
2
x
y
2  2 1
a b
Cartesian Equation for Hyperbola
• The derivation for a hyperbola works
exactly the same, except for the sign
difference, yielding
2
2
x
y
2  2 1
a b
(Foci are at (c, 0), difference between
focal distances is 2a, and c2 = a2 + b2,
eccentricity e = c/a)
Cartesian Equation for Parabola
• If we put the focus at (0, c) and the directrix
as the line y = -c, we get the equation
x  (y  c)  y  c
• Squaring and simplifying, we obtain
x2 = 4cy
• Naturally, we can translate any of these
equations to have center (h, k) by replaceing
x with (x - h) and y with (y - k)
2
2
Info from the Equations
• We should be able to go from descriptions
(“ellipse with its foci at (2, 5) and (2, 1) and
eccentricity 0.4”) to Cartesian equations
(here, center is (2, 3) so c = 2, c/a = 0.4 and
a = 5, and therefore b2 = 52- 22 = 21, giving
the equation (x - 2)2/21 +(y - 3)2/25 = 1)
• We should be able to go backward, finding
center, foci, directrices, eccentricity, etc.
from the Cartesian equation
General Quadratic Equation
• The general quadratic equation looks like
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
• With no cross-product term, we could
complete the square and reduce to one of
our standard equations
• So we get rid of the cross-product term
• If we replace x with x’cos  - y’sin  and
replace y with x’sin  + y’cos  we get….
General Quadratic Equations
A’x’2 + B’x’y’ + C’y’2 + D’x’ + E’y’ + F’ = 0
• We can obtain formulas for A’, B’, etc. The
only important one is
B’ = Bcos(2 ) +(C - A)sin(2 )
• So if we set tan(2 ) = B/(A - C), B’ will
equal zero
Rotation of Axes
• This process is called rotation of axes
• The new x’ and y’ axes are at angle  to the
original axes
• We can rotate the axes to eliminate the
cross-product term, then complete the
squares to reduce the equation to standard
form
Example
Tell about 2x2 + 24xy + 9y2 +5x - 10y - 30 = 0
• First, if tan(2 ) = -24/7, cos() = 3/5 and
sin() = 4/5
• So take x = 3x’/5 - 4y’/5, y = 4x’/5 + 3y’/5
• Substitute to obtain
18x’2 - 7y’2 - 5x’ - 10y’ - 30 = 0
Example
18(x’2 - 5/18 x’) - 7(y’2 - 10/7y’) - 30 = 0
18(x’2 - 5/18 x’ + 25/1296) - 7(y’2 - 10/7 y’
+25/49) - 30 - 25/72 + 25/7 = 0
18(x’ - 5/36)2 - 7(y’ - 5/7)2 = 16745/504
Discriminant
• The combination B2 - 4AC is left unchanged
by any rotation of axes
• In particular, with the correct rotation, B = 0
• AC then tells whether you have an ellipse
(both same sign), hyperbola (opposite sign)
or parabola (one is zero)
• B2 - 4AC is called the discriminant. If
negative, we have an ellipse. Positive gives
a hyperbola. Zero gives a parabola.
Degenerate Conics
• If the discriminant is positive, the hyperbola
may degenerate into a pair of crossing lines
• If the discriminant is negative, the ellipse
might be a circle, a single point, or an
empty graph
• If the discriminant is zero, the parabola may
degenerate into a straight line, a pair of
parallel lines, or an empty graph
Asymptotes
• Hyperbolas have asymptotes
2
2
(x  h) (y  k)

 1 the asymptotes
• For
2
2
a
b
are found by setting the right-hand side to 0
• Then we get the lines y - k = (b/a)(x - h)
Parametric Equations
• Using the trig relations cos2  + sin2  = 1
and sec2  - tan2  = 1 we can transform any
conic into a parametric equation
• Let x = h + a cos  and y = k+ b sin . Then
(x - h)/a = cos , (y - k)/b = sin , so we
have the ellipse
(x  h) (y  k)

1
2
2
a
b
2
2
Parametric Equations
• Similarly, a general hyperbola might be
given by x = h + tan , y = k + sec  (for
hyperbolas that open up and down) or by
x = h + sec , y = k + tan  (for hyperbolas
that open to the sides)
• Parabolas can have y expressed as a
function in x, or vice-versa, which are easy
to turn into parametrics—let x (or y) be the
parameter!
Polar Coordinates
• Using the directrix-focus properties of
conics, it is easy to find equations for conics
in polar coordinates.
• Recall x = r cos  and y = r sin 
• Put a focus at the origin and make the
corresponding directrix the line x = p
• Recall that all conics have the distance from
a point to the focus e (eccentricity) times
the distance from the point to the directrix
Polar Coordinates
• So if (x, y) is on a conic we have that r
equals e(x - p)
• Using x = r cos , we get r = e(r cos  - p)
or r(1 - e cos ) = -ep
• Thus the equation becomes
ep
r
(1  e cos)
• We can rotate this to have different
orientations by manipulating 