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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