MTH253MTH253Calculus IIICalculus III

Chapter 10, Part I (sections 10.1 – 10.3)

Conic Sections

ConicsConicsThe intersection of a right circular cone and a plane will produce one of four curves:

• Parabola

• Ellipse

• Circle

• Hyperbola

2 2 0Ax Bxy Cy Dx Ey F

ConicsConics

Not rotated◦B = 0◦Lines of symmetry are horizontal or

verticalNot 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

2 2 0Ax Bxy Cy Dx Ey F

The ParabolaThe Parabola2 0Ax Ey

2 4x py

Vertical Axis of Symmetry

(0, p)

(0,0)

(2p,p)

1e

y p Note: If p < 0, then just flip this upside-down.

0x i.e. the y-axis

The Parabola - ExampleThe Parabola - Example

22 11 0x y

1e

The Parabola - ExampleThe Parabola - ExampleFind the equation of the parabola with its vertex at the origin and directrix the equation y = –5

The ParabolaThe Parabola2 0Cy Dx

2 4y px

Horizontal Axis of Symmetry

(p,0)

(0,0)

(p,2p) 1ex p

Note: If p < 0, then just flip this to the right.

0y i.e. the x-axis

The EllipseThe Ellipse2 2 0Ax Cy F

0AC

2 22 2 2

2 2 1 , x y c a ba b

Horizontal Major Axis

(c,0)

(0,b)

(a,0)

a b

cea

2a ax

e c

The Ellipse - ExampleThe Ellipse - Example

2 24 9 36 0x y

e

The Ellipse - ExampleThe Ellipse - ExampleFind the equation of the ellipse with its foci at (2,0) and eccentricity of 0.25.

The EllipseThe Ellipse2 2 0Ax Cy F

0AC

2 22 2 2

2 2 1 , x y c a bb a

Vertical Major Axis

(0,c)

(0,a)

(b,0)

a b

cea

2a ay

e c

The CircleThe Circle2 2 0Ax Cy F

0A C

2 22 2 2

2 2 1 or x y x y rr r

(0,r)

(r,0)

0e

(0,0)

The HyperbolaThe Hyperbola2 2 0Ax Cy F

0AC

2 22 2 2

2 2 1 , x y c a ba b

Horizontal Focal Axis

(c,0)

(0,b)

(a,0)

cea

2a axe c

by xa

The Hyperbola - ExampleThe Hyperbola - Example

2 24 9 36 0x y

e

The Hyperbola - ExampleThe Hyperbola - ExampleFind the equation of the hyperbola with its vertices at (3,0) and a directrix x = 2.

The HyperbolaThe Hyperbola2 2 0Ax Cy F

0AC

2 22 2 2

2 2 1 , y x c a ba b

Vertical Focal Axis

(0,c)

(b,0)

(0,a)

cea

2a aye c

ay xb

PF = e * PDPF = e * PD

F

P D

F

PD

D

PF

e < 1 e > 1

e = 1

TranslationsTranslationsTo 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.

2 4x h p y k

2 2

2 2 1x h y ka b

2 2

2 2 1x h y ka b

Translations – Translations – “Complete the “Complete the Square”Square”

2 2 0Ax Cy Dx Ey F

2 23 6 12 1 0x y x y

2 10 2 5 0x x y

Examples:

Rotations – Rotations – The “Cross Product The “Cross Product Term”Term”

2 2 0Ax Bxy Cy Dx Ey F

Angle of Rotation

1 012 tan or 45B

A C

' cos 'sinx x y

' sin 'cosy x y

Substitutions

Note: Use the rotations calculator!

The DescriminantThe Descriminant2 2 0Ax Bxy Cy Dx Ey F

2 4B AC

2 4 0 ParabolaB AC

2 4 0 HyperbolaB AC

2 4 0 EllipseB AC