New Chapter 11 Polar Coordinates and Conic Sections(1)
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CHAPTER 11 POLAR COORDINATES AND CONIC SECTION 11.3 Polar Coordinates Coordinate systems are just ways to define a point in space. For instance in the Cartesian coordinate system at point is given the coordinates (x,y) and we use this to define the point by starting at the origin and then moving x units horizontally followed by y units vertically. Instead of moving vertically and horizontally from the origin to get to the point we could instead go straight out of the origin until we hit the point and then determine the angle this line makes with the positive x-axis. We could then use the distance of the point from the origin and the amount we needed to rotate from the positive x-axis as the coordinates of the point. This system is known as the polar coordinate system. It is easier to work with for many problems. In this coordinate system a point in the plane is located in reference to a fixed point O called the origin or pole. Using O as the endpoint, construct an initial ray called the polar axis. Each point P in the polar coordinate system is represented by an ordered pair (r, ). This ordered pair (r, ) is referred to as the polar coordinates of P. r is the directed distance from O to P and the directed angle from the polar axis to the line OP. A point P can be located as follows: (a) Start at the polar axis and rotate through an angle of measure to determine the ray . (b) On the ray move r units from O to locate P. MAT133 Calculus with Analytic Geometry II Page 89
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Polar Coordinates and Conic Sections
Transcript of New Chapter 11 Polar Coordinates and Conic Sections(1)
PARABOLA
CHAPTER 11 POLAR COORDINATES AND CONIC SECTION11.3Polar Coordinates
Coordinate systems are just ways to define a point in space. For instance in the Cartesian coordinate system at point is given the coordinates (x,y) and we use this to define the point by starting at the origin and then moving x units horizontally followed by y units vertically. Instead of moving vertically and horizontally from the origin to get to the point we could instead go straight out of the origin until we hit the point and then determine the angle this line makes with the positive x-axis. We could then use the distance of the point from the origin and the amount we needed to rotate from the positive x-axis as the coordinates of the point. This system is known as the polar coordinate system. It is easier to work with for many problems. In this coordinate system a point in the plane is located in reference to a fixed point O called the origin or pole. Using O as the endpoint, construct an initial ray called the polar axis. Each point P in the polar coordinate system is represented by an ordered pair (r,
). This ordered pair (r, ) is referred to as the polar coordinates of P. r is the directed distance from O to P and the directed angle from the polar axis to the line OP.
A point P can be located as follows:(a)
Start at the polar axis and rotate through an angle of measure to determine the ray .(b)On the ray move r units from O to locate P.
Example 11.1:Plot the following points in the polar coordinate system.
(a)
(b)
(c)
(d)
This leads to an important difference between Cartesian coordinates and polar coordinates. In Cartesian coordinates there is exactly one set of coordinates for any given point. This means that each point (x, y) has a unique representation. For the polar coordinates this isnt true. In polar coordinates there is literally an infinite number of coordinates for a given point.
Relationship between polar and rectangular coordinates.
x = r cos
y = r sin
A polar equation is an equation whose variables are r and . To convert a polar equation in and r to a rectangular equation in x and y, replace r cos by x, r sin by y and by .
Example 11.2: Replace the following polar equations by equivalent Cartesian equations(a)
Express r = 2sin in rectangular coordinates.
(b)Show that is an equation of a parabola.
To convert a rectangular equation in x and y to a polar equation in rand (, replace x by and y by .
Example 11.1.3: Convert each rectangular equation to a polar equation
(a)
(b)
Homework
Exercise 11.3: 27, 29, 31, 33, 35, 39, 41, 53, 55, 57, 59, 6311.4Graphing in Polar Coordinates
This section describes techniques for graphing equations in polar coordinates.
Graphs of Polar Coordinates
(a)The graph of a polar equation is the set of all points whose polar coordinates satisfy the equation.(b)The method for graphing a polar equation is the point-plotting method.
Step 1: Create the table of values that satisfy the equation.Step 2:Plot the ordered pairs.Step 3:Connect the points with a smooth curve.
Example 11.4.1: Graph each of the following polar equations.
(a)
(b)
(c)
(d)
(e)
Common Polar Coordinate Graphs
Lets identify a few of the more common graphs in polar coordinates. Well also take a look at a
couple of special polar graphs.
Lines
1.
This is a line that goes through the origin and makes an angle of with the positive x-axis. Or, in other words it is a line through the origin with slope of tan .
2.
This is easy enough to convert to Cartesian coordinates to x = a. This is a vertical line.
3.
This converts to y = b and so is a horizontal line.Circles1.
r = a .
This equation is saying that no matter what angle weve got the distance from the origin must be a. This is the definition of a circle of radius a centered at the origin.
2. This is a circle of radius a and center (a,0).
3.This is a circle of radius b and center (0,b) .Cardioids and These have a graph that is vaguely heart shaped and always contain the origin.
11.5Areas and Lengths in Polar Coordinates
Area of a Polar Region
The area problem in Polar Coordinates: Find the area of the region R between a polar curve r = f() and two lines, .
The development of the formula for the area of a polar region parallels that for the area of a region on the rectangular coordinate system. In the polar coordinate system sectors of a circle is used instead of rectangles.
If ) is continuous and non-negative for , then the area A enclosed by the polar curve and the lines is given by
Steps
1.Sketch the region R whose area is to be determined.
2.Draw an arbitrary radical line from the origin to the boundary of the curve.
3.Over what interval of values must vary in order for the radical line to sweep out the region A.
4.The answer in step 3 will determine the lower and upper limits of integration.
Example 11.5.1: Find the area of the region in the first quadrant within the cardiod
In this case we can use the above formula to find the area enclosed by both and then the actual area is the difference between the two.Example 11.5.2: Find the area of the region that is inside the cardiod r = 4 + 4cos and outside the circle r = 6.
Example 11.5.3: Find the area of the region outside the cardiod r = 1 + cos and inside the circle r =sin .
Homework
Exercise 11.5: 9, 11, 12, 14, 15, 1711.6Conic Sections
Objective
Know the names of the conics
The curves that can be obtained by intersecting a cone with a plane are called conics or conic sections. The most important of the conic sections are the circles, the ellipses, the parabolas and the hyperbolas.
A circle is obtained by intersecting a cone with a plane which is perpendicular to the axis and does not contain the vertex.
If the plane is tilted slightly the resulting intersection is an ellipse.
A plane which is tilted further the resulting intersection is a parabola.
If the plane is parallel to the axis but does not contain the vertex, the resulting intersection is a hyperbola.
The study of the conic sections dates back to the ancient Greek geometers. The work was purely geometric and the algebraic formulations were not introduced until the seventeenth century. The four curves have played a vital role in mathematics and its applications. Kepler discovered that the planets revolve around the sun in elliptic orbits. Today, properties of conic sections are used in the construction of telescopes, radar antennas and navigational systems and in determining satellite orbits.
The Parabola
Objectives
Find the equation of a parabola
Discuss the equation of a parabola
Work with parabolas with vertex at (h, k)
Graph parabolas
A parabola is the set of all points in the plane that are equidistant from a given line and a given point not on the line.
All parabolas are vaguely U shaped and they will have a highest or lowest point that is called the vertex. Every parabola has an axis of symmetry and, the graph to either side of the axis of symmetry is a mirror image of the other side. This means that if we know a point on one side of the parabola we will also know a point on the other side based on the axis of symmetry. Intercepts are the points where the graph will cross the x or y-axis.
Terms
(i)focus: the given point
(ii)directrix: the given line
(iii)axis: the line that passes through the focus at right angles to the directrix. The parabola is symmetric about this line.
(iv)vertex: point of intersection of the parabola and the axis.
Equation of the parabola with the vertex at (h, k)
i.
ii.
iii.
iv.
Example 11.6.1: Find the focus and the directrix of the parabola with equation .
Example 11.6.2: Find an equation for the parabola with vertex (1, 2) and focus (4, 2).
Example 11.6.3: Sketch the parabola and label it completely.
Example 11.6.4: Show that the curve is a parabola. Sketch and label it completely.
Homework
Sketch the parabola and label it completely.
1.
2.
3.
Answer
1.Opens in the positive x-direction
2.Opens in the negative y-direction
Vertex:(2, 3)Vertex:((2, (2)
axis:
axis:
focus:
focus:
directrix:
directrix:
3.
Vertex:
Opens in the negative y-direction
axis:
focus: (2, 2)
directrix:
The Ellipse
Objectives
Find the equation of an ellipse
Discuss the equation of an ellipse
Work with ellipses with center at (h, k)
Graph ellipses
An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points is a constant.
Terms(i)foci: the two fixed points
(ii)center: the midpoint of the line segment connecting the foci
(iii)vertices: points of intersection of the ellipse and the line through the foci
(iv)major axis: line that joins the vertices
(v)minor axis: line that is through the center and perpendicular to the major axis
Equation of the ellipse with center at (h, k):
(i)
major axis is parallel to the x-axis
(ii)
major axis is parallel to the y-axis
Example 11.6.5: Sketch the graph of and label it completely.
Example 11.6.6: Graph the ellipse and label it completely.
Example 11.6.7: Sketch the graph of the equation .Homework
Graph each of the following ellipse and label it completely.
1.
2.
3.
4.
5.
Answers
1.Center:origin
2.Center:origin
Major axis:y-axis
Major axis:x-axis
Minor axis:x-axis
Minor axis:y-axis
Foci:
Foci:
Vertices:
Vertices:
Co-vertices:
Co-vertices:
3.Center:(2, 1)4.
4.Center : ((2, 3)
Major axis:parallel to x-axis
Major axis : parallel to y-axis
Minor axis:
Minor axis:
Foci:
Foci :
Vertices:
Vertices:
Co-vertices:
Co-vertices:
5.
Center:(2, (1)
Major axis:parallel to the x-axis
Minor axis:
Foci:
Vertices:
Co-vertices:
The Hyperbola
Objectives
Find the equation of a hyperbola
Discuss the equation of a hyperbola
Work with hyperbolas with center at (h, k)
Graph ellipses
Find the asymptotes of a hyperbola
A hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points is a given positive constant.
Terms:(i)foci: the two fixed points
(ii)center: the midpoint of the line segment joining the foci
(iii)focal axis: the line through the foci. The focal axis also known as the transverse axis.
(iv)conjugate axis: the line through the center and perpendicular to the focal axis.
(v)vertices: the points of intersection of the hyperbola and the focal axis.
Equation of the hyperbola with center at (h, k)
(i)
focal axis is parallel to the x-axis
(ii)
focal axis is parallel to the y-axis
The hyperbola has asymptotes .
The hyperbola has asymptotes .
Example 11.6.8: Sketch the graph of and label it completely.
Example 11.6.9: Sketch the graph of and label it completely.
Homework
Sketch each of the following hyperbola and label it completely.
1.
2.
3.
4.
Answer
1.Focal axis:x-axis2.
Conjugate axis:y-axis
Focal axis
:y-axis
Asymptotes:
Conjugate axis
:x-axis
Vertices:
Asymptotes
:
Foci:
Vertices
:
Foci
:
3.Center:(2, 4)4.Center: ((2, (3)
Focal axis:
Focal axis:
Conjugate axis:
Conjugate axis:
Asymptotes:
Asymptotes:
Vertices:
Vertices:
;
Foci:
Foci:
Homework
Exercise 11.6: 57, 59, 61, 63, 65,
Practice Exercises:15, 17, 19, 21, 47, 49, 51, 53
A = QUOTE
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