Transcript MTH 251 Differential Calculus

MTH 253
Calculus (Other Topics)
Chapter 10 – Conic Sections and
Polar Coordinates
Section 10.3 – Quadratic Equations
and Rotations
Copyright © 2009 by Ron Wallace, all rights reserved.
Quadratic Equations
Ax  Cy  Dx  Ey  F  0
2
2
 Any equation of the above form can be
changed to the graphing form of a conic
by completing the square.
Quadratic Equations – Example 1
x  2 y  6x  12 y  7  0
2
2
Quadratic Equations – Example 2
3x  6x  12 y  7  0
2
The “Cross Product” Term
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
 The extra term, Bxy is called the “cross
product” term.
 Its affect?
 Causes the conic to be rotated.
 i.e. The axis of symmetry is not parallel to the
x-axis or the y-axis.
 Solution? “Rotation of Axis”
Rotation of Axis
 Find another coordinate system with
the same origin such that the axis of
symmetry of the conic, relative to this
new coordinate system, is parallel to
one of the axis.
y'
y
x'
x
Question: What is the relationship between (x,y) and (x’,y’) for the same point?
Rotation of Axis
x '  r cos 
What is the relationship between
(x,y) and (x’,y’) for the same point?
y'
y '  r sin 
y
(x,y) = (x’,y’)
r


x  r cos(   )  r cos cos   r sin  sin 
x  x 'cos   y 'sin 
y  x 'sin   y 'cos 
That is: Replacing x & y by these expressions will give an
equation/point relative to the new coordinate system.
x'
x
Rotation of Axis – Example 1
x  x 'cos   y 'sin 
y  x 'sin   y 'cos 
If (x,y) = (2,5), find (x’,y’) if  = 30
y'
y
(x,y) = (x’,y’)
r


That is: Replacing x & y by these expressions will give an
equation/point relative to the new coordinate system.
x'
x
Rotation of Axis – Example 2
x  x 'cos   y 'sin 
y'
y  x 'sin   y 'cos 
Given the line y = 2x – 1,
find the equation y’ = mx’ + b if  = 60
y
(x,y) = (x’,y’)
r


That is: Replacing x & y by these expressions will give an
equation/point relative to the new coordinate system.
x'
x
Rotation of Axis – Eliminating Bxy
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
 What angle of rotation must be used
in order to eliminate the Bxy term?
 Substituting the conversion formulas
into this equation gives …
A ' x '  B ' x ' y ' C ' y '  D ' x ' E ' y ' F '  0
2
2
… where …
B '  B cos2  (C  A)sin 2
Rotation of Axis – Eliminating Bxy
 Since we need B’ = 0 …
B '  B cos2  (C  A)sin 2  0
 That is …
 AC 
  cot 

 B 
1
2
1
If A = C,  = 45
or
 B 
  tan 

 AC 
1
2
1
Use this one if A  C
Rotation of Axis – Eliminating Bxy
 Using …
1
B 
1 
tan
if A  C
2



 AC 
0

45
if A  C

… the new coefficients are …
A '  A cos2   B cos  sin   C sin 2 
B'  0
C '  A sin 2   B cos  sin   C cos 2 
D '  D cos   E sin 
E '   D sin   E cos 
F' F
NO, you do NOT need to memorize these!
Which conic is it?
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
 The Discriminant Test
 Parabola if
B 2  4 AC  0
 Ellipse if
B 2  4 AC  0
 Hyperbola if B 2  4 AC  0
NOTE: For any rotation, B
2
 4 AC  B '2  4 A ' C '