Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

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Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations

Section 10.1 Geometry of Parabola, Ellipse, Hyperbola

a. Geometric Definition

b. Parabola

c. Ellipse

d. Hyperbola

e. Translations

f. Distance Between a Point and Line

g. Parabolic Mirrors

h. Optical Consequences

i. Elliptical Reflectors

j. Hyperbolic Reflectors Section 10.2 Polar Coordinates

a. Illustrative Figure

b. Assigning Polar Coordinates

c. Properties 1 and 2

d. Property 3

e. Relation to Rectangular Coordinates

f. Properties Relating Polar and Rectangular Coordinates

g. Simple Sets

h. Symmetry

Section 10.3 Sketching Curves in Polar Coordinates

a. Spiral of Archimedes

b. Example

c. Lines

d. Circles

e. Limaçons

f. Lemniscates

g. Petal Curves

h. Intersection of Polar Curves

Section 10.4 Area in Polar Coordinates

a. Computing Area

b. Properties

Section 10.5 Curves Given Parametrically

a. Parameterized Curve

b. Straight Lines

c. Ellipses and Circles

d. Hyperbolas

Section 10.6 Tangents to Curves Given Parametrically

a. Assumptions

b. Properties

Section 10.7 Arc Length and Speed

a. Length of a Curve

b. Formula

c. Length of the Graph of f

d. Geometric Significance of dx/ds and dy/ds

e. Speed Along a Plane Curve

Section 10.7 The Area of a Surface of Revolution; The Centroid of a Curve; Pappus’s Theorem on Surface Area

a. The Area of a Surface of Revolution

b. Computing Area

c. Centroid of a Curve

d. Formulas

e. Pappus’s Theorem on Surface Area

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Geometry Of Parabolas

Geometric Definition

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Geometry Of Parabolas

Parabola

Standard Position F on the positive y-axis, l horizontal. Then F has coordinates of the form (c, 0) with c > 0 and l has equation x = −c.

Derivation of the Equation A point P(x, y) lies on the parabola iff d 1 = d 2 , which here means

This equation simplifies to

x 2 = 4cy.

Terminology A parabola has a focus, a directrix, a vertex, and an axis.

( ) 2

x 2 + yc = + y c

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Geometry Of Ellipses

Ellipse

Standard Position F 1 and F 2 on the x-axis at equal distances c from the origin.

Then F 1 is at (−c, 0) and F 2 at (c, 0). With d 1 and d 2 as in the defining figure, set d 1 + d 2 = 2a.

Equation

Setting , we have

2 2

2 2 2 1

x y

a + a c =

2 2

b = ac

2 2

2 2 1

x y

a + b =

Terminology An ellipse has two foci, F 1 and F 2 , a major axis, a minor axis, and four vertices. The point at which the axes of the ellipse intersect is called the center of the

ellipse.

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Geometry Of Hyperbolas

Hyperbola

Standard Position F 1 and F 2 on the x-axis at equal distances c from the origin.

Then F 1 is at (−c, 0) and F 2 at (c, 0). With d 1 and d 2 as in the defining figure, set

|d 1 − d 2 | = 2a Equation

Terminology A hyperbola has two foci, F 1 and F 2 , two vertices, a transverse axis that joins the two vertices, and two asymptotes. The midpoint of the transverse axis is called the center of the hyperbola.

Setting , we have

2 2

2 2 2 1

x y

ac a =

2 2

b = ca

2 2

2 2 1

x y

ab =

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Translations

Suppose that x 0 and y 0 are real numbers and S is a set in the xy-plane. By replacing each point (x, y) of S by (x + x 0 , y + y 0 ), we obtain a set S ´ which is congruent to S and obtained from S without any rotation. Such a displacement is called a translation.

The translation

(x, y ) → (x + x 0 , y + y 0 )

applied to a curve C with equation E(x, y) = 0 results in a curve C ´ with equation E(x − x 0 , y − y 0 ) = 0.

Geometry Of Parabola, Ellipse, Hyperbola

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The distance between the origin and any line l : Ax + By + C = 0 is given by the formula

By means of a translation we can show that the distance between any point P(x 0 , y 0 ) and the line l : Ax + By + C = 0 is given by the formula

( ) 0, 2 C 2

d l

A B

= +

Geometry Of Parabola, Ellipse, Hyperbola

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Parabolic Mirrors

Take a parabola and revolve it about its axis. This gives you a parabolic surface. A curved mirror of this form is called a parabolic mirror. Such mirrors are used in searchlights (automotive headlights, flashlights, etc.) and in reflecting telescopes.

Geometry Of Parabola, Ellipse, Hyperbola

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Geometry Of Parabola, Ellipse, Hyperbola

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Elliptical Reflectors

Geometry Of Parabola, Ellipse, Hyperbola

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Hyperbolic Reflectors

Geometry Of Parabola, Ellipse, Hyperbola

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Polar Coordinates

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Polar Coordinates

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Polar Coordinates

Polar coordinates are not unique. Many pairs [ r, θ] can represent the same point.

(1) If r = 0, it does not matter how we choose θ. The resulting point is still the pole:

(2) Geometrically there is no distinction between angles that differ by an

integer multiple of 2 π. Consequently:

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Polar Coordinates

(3) Adding π to the second coordinate is equivalent to changing the sign of the

first coordinate:

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Polar Coordinates

Relation to Rectangular Coordinates

The relation between polar coordinates [ r, θ] and rectangular coordinates

(x, y) is given by the following equations:

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Polar Coordinates

Unless x = 0,

and, under all circumstances,

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Polar Coordinates

Here are some simple sets specified in polar coordinates.

(1) The circle of radius a centered at the origin is given by the equation r = a.

The interior of the circle is given by r < a and the exterior by r > a.

(2) The line that passes through the origin with an inclination of α radians has polar equation

θ = α.

(3) For a ≠ 0, the vertical line x = a has polar equation

r cos θ = a or, equivalently, r = a sec θ (4) For b ≠ 0, the horizontal line y = b has polar equation

r sin θ = b or, equivalently, r = b csc θ.

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Polar Coordinates

Symmetry

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Sketching Curves in Polar Coordinates

Example

Sketch the curve r = θ, θ ≥ 0 in polar coordinates.

Solution

At θ = 0, r = 0; at θ = ¼π, r = ¼ π; at θ = ½π, r = ½ π; and so on. The

curve is shown in detail from θ = 0 to θ = 2π in Figure 10.3.1. It is an unending

spiral, the spiral of Archimedes. More of the spiral is shown on a smaller scale

in the right part of the figure.

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Sketching Curves in Polar Coordinates

Example

Sketch the curve r = cos 2 θ in polar coordinates.

Solution

Since the cosine function has period 2 π, the function

r = cos 2 θ has period π. Thus it may seem that we can restrict ourselves to sketching the curve from θ = 0 to θ = π. But this is not the case. To obtain the complete curve, we must

account for r in every direction; that is, from θ = 0 to θ = 2π.

Translating Figure 10.3.4 into polar coordinates [ r, θ], we obtain a sketch of the curve r = cos 2θ

in polar coordinates (Figure 10.3.5). The sketch is developed in eight stages.

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Sketching Curves in Polar Coordinates

Lines : θ = a, r = a sec θ, r = a csc θ.

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Sketching Curves in Polar Coordinates

Circles : r = a, r = a sin θ, r = a cos θ.

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Sketching Curves in Polar Coordinates

Limaçons : r = a + b sin θ, r = a + b cos θ.

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Sketching Curves in Polar Coordinates

Lemniscates: r² = a sin 2 θ, r² = a cos 2θ

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Sketching Curves in Polar Coordinates

Petal Curves: r = a sin nθ, r = a cos nθ, integer n.

If n is odd, there are n petals. If n is even, there are 2n petals.

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Sketching Curves in Polar Coordinates

The Intersection of Polar Curves

The fact that a single point has many pairs of polar coordinates can cause

complications. In particular, it means that a point [r 1 , θ 1 ] can lie on a curve given by a polar equation although the coordinates r 1 and θ 1 do not satisfy the equation. For example, the coordinates of [2 , π] do not satisfy the equation r 2 = 4 cos θ:

r 2 = 2 2 = 4 but 4 cos θ = 4 cos π = −4.

Nevertheless the point [2 , π] does lie on the curve r 2 = 4 cos θ. We know this because [2 , π] = [−2, 0] and the coordinates of [−2, 0] do satisfy the equation:

r 2 = (−2) 2 = 4, 4 cos θ = 4 cos 0 = 4

In general, a point P[r 1 , θ 1 ] lies on a curve given by a polar equation if it has at least one polar coordinate representation [ r, θ] with coordinates that satisfy the equation.

The difficulties are compounded when we deal with two or more curves.

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Area in Polar Coordinates

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Area in Polar Coordinates

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Curves given Parametrically

Assume a pair of functions x = x(t), y = y(t) is differentiable on the interior of an interval I. At the endpoints of I (if any) we require only one-sided continuity.

For each number t in I we can interpret (x(t), y(t)) as the point with x-coordinate

x(t) and y-coordinate y(t). Then, as t ranges over I, the point (x(t), y(t)) traces out a

path in the xy-plane. We call such a path a parametrized curve and refer to t as the

parameter.

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Curves given Parametrically

Straight Lines

Given that (x 0 , y 0 ) = (x 1 , y 1 ), the functions

parametrize the line that passes through the points (x 0 , y 0 ) and (x 1 , y 1 ).

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Curves given Parametrically

Ellipses and Circles

Usually we let t range from 0 to 2 π and parametrize the ellipse by setting

If b = a, we have a circle. We can parametrize the circle x 2 + y 2 = a 2

by setting

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Curves given Parametrically

Hyperbolas

Take a, b > 0. The functions x(t) = a cosh t, y(t) = b sinh t satisfy the identity

Since x(t) = a cosh t > 0 for all t, as t ranges over the set of real numbers, the point (x(t), y(t)) traces out the right branch of the hyperbola

( ) 2 ( ) 2

2 2 1

x t y t

a b

   

  −   =

2 2

2 2 1

x y

ab =

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Tangents to Curves Given Parametrically

Let C be a curve parametrized by the functions x = x(t), y = y(t)

defined on some interval I. We will assume that I is an open interval and the parametrizing functions are differentiable.

Since a parametrized curve can intersect itself, at a point of C there can be (i) one tangent, (ii) two or more tangents, or (iii) no tangent at all.

To make sure that there is at least one tangent line at each point of C, we will make

the additional assumption that

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Tangents to Curves Given Parametrically

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Arc Length and Speed

Figure 10.7.1 represents a curve C parametrized by a pair of functions x = x(t), y = y(t) t ∈ [a, b].

We will assume that the functions are continuously differentiable on [a, b] (have first derivatives which are continuous on [a, b]). We want to determine the

length of C.

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Arc Length and Speed

The length of the path C traced out by a pair of continuously differentiable functions

x = x(t), y = y(t) t ∈ [a, b]

is given by the formula

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Arc Length and Speed

Suppose now that C is the graph of a continuously differentiable function y = f (x), x ∈ [a, b].

We can parametrize C by setting

x(t) = t, y(t) = f (t) t ∈ [a, b].

Since

x ´ (t) = 1 and y ´ (t) = f ´ (t), (10.7.1) gives

( ) b 1 ( ) 2

L C = ∫ a +   ft   dt

Replacing t by x, we can write:

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Arc Length and Speed

The Geometric Significance of dx/ds and dy/ds

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Arc Length and Speed

Speed Along a Plane Curve

So far we have talked about speed only in connection with straight-line motion.

How can we calculate the speed of an object that moves along a curve? Imagine an object moving along some curved path. Suppose that (x(t), y(t)) gives the position of the object at time t. The distance traveled by the object from time zero to any later time t is simply the length of the path up to time t:

The time rate of change of this distance is what we call the speed of the object.

Denoting the speed of the object at time t by ν(t), we have

( ) ( ) 2 ( ) 2

0

s t = ∫ t   x u ′  +    y u ′   du

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The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area

The Area of a Surface of Revolution

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The Area Of A Surface Of Revolution; The Centroid

Of A Curve; Pappus's Theorem On Surface Area

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The Area Of A Surface Of Revolution; The Centroid Of A Curve; Pappus's Theorem On Surface Area

Centroid of a Curve

We can locate the centroid of a curve from the following principles, which we take from physics.

Principle 1: Symmetry. If a curve has an axis of symmetry, then the centroid lies somewhere along that axis.

Principle 2: Additivity. If a curve with length L is broken up into a finite number of pieces with arc lengths Δs 1 , . . . , Δs n and centroids then

( x y , )

( x y 1 , 1 ) ,  , ( x y n , n ) ,

1 1 n n and 1 1 n n

xL = ∆ + x s  + ∆ x s yL = ∆ + y s  + ∆ y s

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The Area Of A Surface Of Revolution; The Centroid

Of A Curve; Pappus's Theorem On Surface Area

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The Area Of A Surface Of Revolution; The Centroid

Of A Curve; Pappus's Theorem On Surface Area

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