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Lines

Precalculus 10.1

Inclination of a Line

Every non-horizontal line must intersect the x-axis. The angle formed by such an intersection determines the inclination of the line, as specified in the following definition.

Horizontal Line Vertical Line Acute Angle Obtuse AngleFigure 10.1

Inclination of a Line Example 1

Find the inclination of the line 0333 yx

Example 2

Find the inclination of the line 035 yx

The Angle Between Two Lines

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Example 3

Find the angle between the two lines:

2

42

yx

yx

The Distance Between a Point and a Line

Finding the distance between a line and a point not on the line is an application of perpendicular lines.

This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 10.6.

Figure 10.6

The Distance Between a Point and a Line

Remember that the values of A, B, and C in this

distance formula correspond to the general

equation of a line, Ax + By + C = 0.

Example 4

Find the distance between the point (0,2) and the line 034 yx

Example 5

Consider the triangle with vertices A (0,0),

B (1,5), and C (3,1).

a) Find the altitude from vertex B to side AC.

b) Find the area of the triangle.

Introduction to Conics: Parabolas

Precalculus 10.2

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Conics

A conic section (or simply conic) is the intersection of a in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone.

Circle Ellipse Parabola Hyperbola

Basic ConicsFigure 10.9

Conics

When the plane does pass through the vertex, the resulting figure is a degenerate conic, as shown in Figure 10.10.

Point Line Two Intersecting Lines

Degenerate Conics

Figure 10.10

Conics

There are several ways to approach the study of conics. You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general second-degree equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

However, you will study a third approach, in which each of the conics is defined as a locus (collection) of points satisfying a geometric property.

Conics

For example, a circle is defined as the collection of all points

(x, y) that are equidistant from a fixed point (h, k).

This leads to the standard form of the equation of a circle

(x – h)2 + (y – k)2 = r 2. Equation of circle

Parabolas

We have learned that the graph of the quadratic function

f (x) = ax2 + bx + c

is a parabola that opens upward or downward.

The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola.

Parabolas

The midpoint between the focus and the directrix is called the vertex, and the line passing through the focus and the vertex is called the axis of the parabola.

Note in Figure 10.11 that a parabola is symmetric with respect to its axis.

Parabola

Figure 10.11

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Parabolas

Using the definition of a parabola, you can derive the following standard form of the equation of a parabolawhose directrix is parallel to the x-axis or to the y-axis.

Parabolas

(a) (x – h)2 = 4p(y – k)Vertical axis: p 0

(b) (x – h)2 = 4p(y – k) Vertical axis: p 0

(c) (y – k )2 = 4p(x – h)Horizontal axis: p 0

(d) (y – k )2 = 4p(x – h)Horizontal axis: p 0

Figure 10.12

Example 1

Find the standard equation of the parabola with vertex (3,2) and focus (1,2).

Example 2

Find the focus of the parabola given by

242

61 xxy

Example 3

Find the standard form of the equation of the parabola with vertex (1,3) and focus (1,5). Then write the quadratic form of the equations.

Ellipses

Precalculus 10.3

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Introduction

The second type of conic is called an ellipse, and is defined as

follows.

d1 + d2 is constant.

Figure 10.19

Introduction

The line through the foci intersects the ellipse at two points

called vertices.

The chord joining the vertices is the major axis, and its

midpoint is the center of the ellipse.

The chord perpendicular to the

major axis at the center is the

minor axis of the ellipse.

See Figure 10.20.

Figure 10.20

Introduction

We can visualize the definition of an ellipse by imagining two thumbtacks placed at the foci, as shown in Figure 10.21.

If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse.

Figure 10.21

Introduction

To derive the standard form of the equation of an ellipse, consider the ellipse in Figure 10.22 with the following points: center, (h, k); vertices, (h a, k); foci, (h c, k).

Note that the center is the

midpoint of the segment

joining the foci.

The sum of the distances

from any point on the ellipse

to the two foci is constant.Figure 10.22

Introduction Introduction

Figure 10.23 shows both the horizontal and vertical orientations for an ellipse.

Figure 10.23

Major axis is horizontal. Major axis is vertical.

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Example 1

Find the standard form of the equation of the ellipse having foci at (-2,2) and (4,2) and a major axis of length 10.

Example 2

Sketch the graph of the ellipse given by

0483649 22 yxyx

Example 3

Find the center, vertices, and foci of the ellipse given by 01650322516 22 yxyx

An Application Involving an Elliptical Orbit

The moon travels about Earth in an elliptical orbit with Earth at

one focus, as shown in Figure 10.27. The major and minor axes

of the orbit have lengths of 768,800 kilometers and 767,640

kilometers, respectively. Find the greatest and smallest

distances (the apogee and perigee, respectively) from Earth’s

center to the moon’s center.

Figure 10.27

Example 4

The first artificial satellite to orbit the Earth was Sputnik I (launched by Russia in 1957). Its orbit was elliptical with the center of Earth at one focus. The major and minor axes of the orbit had lengths of 13,906 kilometers and 13,887 kilometers, respectively. Find the apogee and perigee (greatest and smallest distances) from Earth’s center to the satellite.

Example 4 cont’d

Use the apogee and perigee to find the least distance and greatest distance of the satellite from Earth’s surface in this orbit (Earth has a radius of 6378 kilometers).

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Eccentricity

One of the reasons it was difficult for early astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetary orbits are relatively close to theircenters, and so the orbits are nearly circular.

To measure the ovalness of an ellipse, you can use the concept of eccentricity.

Note that 0 < e < 1 for every ellipse.

Example 5

Find the eccentricity of the ellipse 18116

22

yx

Hyperbolas

Precalculus 10.4

Introduction

Figure 10.30

d2 – d1 is a positive constant.

Introduction

The graph of a hyperbola has two disconnected branches.

The line through the two foci intersects the hyperbola at its two vertices.

The line segment connecting

the vertices is the transverse

axis, and the midpoint of the

transverse axis is the center

of the hyperbola.

See Figure 10.31.Figure 10.31

Introduction

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Introduction

Figure 10.32 shows both the horizontal and vertical orientations for a hyperbola.

Transverse axis is horizontal. Transverse axis is vertical.

Figure 10.32

Example 1

Find the standard form of the equation of the hyperbola with foci (0,0) and (8,0) and vertices (3,0) and (5,0).

Asymptotes of a Hyperbola

Each hyperbola has two asymptotes that intersect at the center of the hyperbola, as shown in Figure 10.34.

The asymptotes pass through the vertices of a rectangle of dimensions 2a by 2b, with its center at (h, k).

The line segment of length 2b joining (h, k + b) and (h, k – b) [or (h + b, k) and (h – b, k)] is the conjugate axis of the hyperbola.

Figure 10.34

Asymptotes of a Hyperbola

Example 2

Sketch the hyperbola whose equation is

3694 22 xy

Example 3

Sketch the hyperbola given by

and find the equation, its asymptotes, and the foci.

040849 22 yyx

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Example 4

Find the standard form of the equation of the hyperbola having vertices (3,2) and (9,2) and having asymptotes and xy

322 .6

32 xy

Example 5 – An Application Involving Hyperbolas

Two microphones, 1 mile apart, record an explosion. Microphone A receives the sound 2 seconds before microphone B. Where did the explosion occur? (Assume sound travels at 1100 feet per second.)

Solution:Assuming sound travels at

1100 feet per second, you know that the explosion took place 2200 feet farther from B than from A, as shown in Figure 10.42.

2c = 5280

2200 + 2(c – a) = 5280Figure 10.42

Example 5 – Solution

The locus of all points that are 2200 feet closer to A than to B is one branch of the hyperbola

where

and

cont’d Example 5 – Solution

So, b2 = c2 – a2

= 26402 – 11002

= 5,759,600,

and you can conclude that the explosion occurred somewhere on the right branch of the hyperbola

cont’d

Applications

Another interesting application of conic sections involves the orbits of comets in our solar system.

Of the 610 comets identified prior to 1970, 245 have elliptical orbits, 295 have parabolic orbits, and 70 have hyperbolic orbits.

The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in Figure 10.43.

Figure 10.43

Applications

Undoubtedly, there have been many comets with parabolic or hyperbolic orbits that were not identified. We only get to see such comets once.

Comets with elliptical orbits, such as Halley’s

comet, are the only ones that remain in our

solar system.

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General Equations of Conics Example 6

Classify the graph of each equation.

a)

b)

c)

d)

0143 22 xyx

02221022 yxyx

042442 yxx

0104122832 22 yxyx

Polar Coordinates

Precalculus 10.7

Introduction

To form the polar coordinate system in the plane, fix a point O, called the pole (or origin), and construct from O an initial ray called the polar axis, as shown in Figure 10.59.

Then each point P in the plane can be assigned polar coordinates (r, ) as follows.

1. r = directed distance from O to P

2. = directed angle, counterclockwise from polar axis to segment

Figure 10.59

Example 1

Plot the given point in the polar coordinate system.

a) b) c) 3,3

6,2

6,2

Example 2

Plot the point and find three additional polar representations of this point, using

43,2

.22

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Coordinate Conversion

To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the origin, as shown in Figure 10.64.

Because (x, y) lies on a circle

of radius r, it follows that

r 2 = x2 + y2.

Figure 10.64

Coordinate Conversion

Moreover, for r > 0, the definitions of the trigonometric functions imply that

If r < 0, you can show that the same relationships hold.

Example 3

Convert each point to rectangular coordinates.

a) b) 63,4

6,2

Example 4

Convert each point to polar coordinates.

a) b) 2,2 0,1

Example 5

Describe the graph of each polar equation and find the corresponding rectangular equation.

a) b) c)1r 4 cscr Graphs of Polar Equations

Precalculus 10.8

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Example 1

Sketch the graph of the polar equation

.sin4 r

Symmetry

Symmetry with respect to the line = /2 is one of three important types of symmetry to consider in polar curve sketching. (See Figure 10.72.)

Symmetry with Respect to

the Line = Symmetry with Respect to the Polar Axis

Symmetry with Respect to the Pole

Figure 10.72

Symmetry Symmetry

Example 2

Use symmetry to sketch the graph of

.sin42 r

Example 3

Sketch the graph of .cos23 r

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Example 4

Sketch the graph of .2cos4 r

Example 5

Sketch the graph of .3cos3 r

Example 6

Sketch the graph of .2cos42 r