MTH253
Calculus III
Chapter 10, Part I
(sections 10.1 – 10.3)
Conic
Sections
Conics
The intersection of a
right circular cone and
a plane will produce
one of four curves:
• Parabola
• Ellipse
• Circle
• Hyperbola
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
Conics
Ax  Bxy  Cy  Dx  Ey  F  0
2

2
Not rotated
◦B=0
◦ Lines of symmetry are horizontal or vertical

Not translated
◦ Parabola: C, D, & F = 0 or A, E, & F = 0
 Vertex at the origin
◦ Ellipse & Hyperbola: D = 0 & E = 0
 “Centered” at the origin
Ax  Ey  0
2
The Parabola
Vertical Axis of Symmetry
x0
i.e. the y-axis
x  4 py
2
e 1
(0, p)
(2p,p)
(0,0)
y  p
Note: If p < 0, then just
flip this upside-down.
The Parabola - Example
2 x  11y  0
2
e 1
The Parabola - Example
Find the equation of the parabola with its vertex
at the origin and directrix the equation y = –5
Cy  Dx  0
2
The Parabola
Horizontal Axis of Symmetry y  0
i.e. the x-axis
y  4 px
2
x  p
(p,2p)
e 1
(p,0)
(0,0)
Note: If p < 0, then just
flip this to the right.
Ax  Cy  F  0
2
The Ellipse
2
AC  0
Horizontal Major Axis
a b
x2 y 2
2
2
2
 2 1 , c  a b
2
a
b
(0,b)
a a2
x 
e c
(c,0) (a,0)
c
e
a
The Ellipse - Example
4 x  9 y  36  0
2
2
e
The Ellipse - Example
Find the equation of the ellipse with its foci at
(2,0) and eccentricity of 0.25.
Ax  Cy  F  0
2
The Ellipse
Vertical Major Axis
2
AC  0
a b
x2 y 2
2
2
2
 2 1 , c  a b
2
b
a
a a2
y 
e c
(0,a)
(0,c)
(b,0)
c
e
a
Ax  Cy  F  0
2
The Circle
2
AC 0
2
x
y
 2 1
2
r
r
2
or
x y r
2
2
2
(0,r)
e0
(0,0)
(r,0)
Ax  Cy  F  0
2
The Hyperbola
2
AC  0
Horizontal Focal Axis
x2 y 2
2
2
2
 2 1 , c  a b
2
a
b
b
y x
a
(0,b)
(a,0)
(c,0)
c
e
a
a
a2
x 
e
c
The Hyperbola - Example
4 x  9 y  36  0
2
2
e
The Hyperbola - Example
Find the equation of the hyperbola with its
vertices at (3,0) and a directrix x = 2.
Ax  Cy  F  0
2
The Hyperbola
2
AC  0
Vertical Focal Axis
y 2 x2
2
2
2
 2 1 , c  a b
2
a b
(0,c)
a
y x
b
(0,a)
(b,0)
c
e
a
a
a2
y 
e
c
PF = e * PD
D
P
D
F
P
F
e<1
e>1
P
F
D
e=1
Translations
To move the center of an ellipse or
hyperbola or the vertex of a parabola to
the point (h, k), replace x with x-h and y
with y-k.
 Treat (h, k) as if it was the origin.

 x  h
 x  h
2
 4p y  k
a
2
 x  h
a
2
2
y k


2
y k


2
b
2
b
1
2
2
1
Translations – “Complete the Square”
Ax  Cy  Dx  Ey  F  0
2
2
Examples:
x  3 y  6 x  12 y  1  0
2
2
x  10 x  2 y  5  0
2
Rotations – The “Cross Product Term”
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
Angle of Rotation
 B 
0
  tan 
or
45

 AC 
1
2
1
Substitutions
x  x ' cos   y 'sin 
y  x 'sin   y ' cos 
Note: Use the
rotations calculator!
The Descriminant
B  4 AC
2
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
B 2  4 AC  0

Parabola
B  4 AC  0

Ellipse
B 2  4 AC  0

Hyperbola
2