CHAPTER 10

CONICS AND POLAR

COORDINATES

10.1 The Parabola• In a plane with line, l, (directrix) and fixed point

F (focus), eccentricity is defined as the ratio of the distance from any point, P, to the focus to the distance to the directrix.

hyperbolaeellipsee

parabolaePL

PFe

:1;:10

;:1,

Parbola, e=1

• Set of point P such that the distance from a point to the focus = distance from point to the directirx.

• Standard equation of a parabola

)(4

)(42

2

verticalpyx

horizontalpxy

10.2 Ellipses & Hyperbolas

1:

10:

ehyperbola

eellipse

PL

PFe

Standard Equation

downupb

x

a

y

rightleftb

y

a

xhyperbola

b

y

a

xellipse

1

1:

1:

2

2

2

2

2

2

2

2

2

2

2

2

10.3 Translation & Rotation of Axes• Conics need not be centered at the origin.

They could be centered at any point: (h,k)

• Let u= x – h, v = y – k

• Equivalently: x = u + h, y = v + k

• May need to complete the square to create standard form for recognition of conic.

Rotation of Axes

• The xy-axes may be rotated through angle theta for any conic

• How is the angle, theta, found?

cossin

sincos

vuy

vux

B

CA

FEyDxCyBxyAxfor

)2cot(

0: 22

10.4 Parametric Representation of Curves in the Plane

• For a parametric function, x=f(t), y=g(t)

• Values of t as t advances from a to b, define where the curve begins and ends

Differentiation of parametric equations

Let f & g be continuously differentiable with f’(t) not equal 0 on a<t<b. Then x=f(t) and y=g(t)

The derivative of y with respect to x is:

dtdxdtdy

dx

dy

Calculation of arc length

dtdt

dy

dt

dxL

22

10.5 Polar Coordinate System

• Given a fixed point (O), the pole or origin, a polar axis running horizontally to the right of the origin, any point can be defined as: distance, r, from the origin, rotated through an angle, theta, from the polar axis.

• The coordinates of the point are of the form: (r, theta)

Relationships between polar & cartesian coordinates

• Polar to Cartesian

• Cartesian to Polar

x

y

yxr

ry

rx

tan

sin

cos

222

Example: Show that the given polar equation is that of an ellipse:

ellipseyxx

xxyx

xyx

xyx

rr

r

r

36161215

1236)(16

64

64

6cos4

6)cos4(cos4

6

22

222

22

22

Polar form of conics

• If a conic has its focus at the pole and its directrix d units away, the final form is:

tyeccentricie

e

edr

o

)cos(1

10.6 Graphs of Polar Equations

• Common polar graphs:– Cardiods– Limacons– Lemniscates– Roses– Spirals

Limacons & Cardiods• If a=b, cardiod

(heart-shaped)

• If a<b, inner loop

sin,cos barbar

Lemniscates• Figure-8 shaped curves

2sin4:

2sin,2cos2

22

rexample

arar

Roses• Polar equations of

the form:

• n leaves (n odd)

• 2n leave (n even)

)3cos(6:

)sin(),cos(

rexample

narnar

Spiral of Archimedes and Logarithmic Spiral

5:

:log

:

rexample

aer

spiralarithmic

ar

Archimedesofspiral

b

10.7 Calculus in Polar Coordinates

• Area in polar coordinates

• Tangents in polar coordinates

Area in polar coordinates

• Recall how to find area of sector of a circle:

• For a polar curve, r = f(theta) and the angle changes as you move along the curve from a to b.

• The area is the sum of all the areas of each little sector, which is an integral:

2

2

1rA

dfA 2)]([2

1

Tangents in Polar Coordinates• In Cartesian coordinates, m = dy/dx

tancos)('

sin)(',,

0)('0)(,,

cos)('sin)(

sin)('cos)(

coscos

sin)(sin:Re

f

fmpoletheatThen

fandfranglesomefor

ff

ff

ddxddy

dx

dy

frx

frycall