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Copyright © 2007 Pearson Education, Inc. Slide 6-2
Chapter 6: Analytic Geometry
6.1 Circles and Parabolas
6.2 Ellipses and Hyperbolas
6.3 Summary of the Conic Sections
6.4 Parametric Equations
Copyright © 2007 Pearson Education, Inc. Slide 6-3
6.1 Circles and Parabolas
• Conic Sections– Parabolas, circles, ellipses, hyperbolas
Copyright © 2007 Pearson Education, Inc. Slide 6-4
• A circle with center (h, k) and radius r has length to some point (x, y)
on the circle. • Squaring both sides yields the center-radius
form of the equation of a circle.
6.1 Circles
A circle is a set of points in a plane that are equidistant from a fixed point. The distance is called the radius of the circle, and the fixed point is called the center.
22 )()( kyhxr
22 )()( kyhxr 2
Copyright © 2007 Pearson Education, Inc. Slide 6-5
6.1 Finding the Equation of a Circle
Example Find the center-radius form of the equation of a circle with radius 6 and center (–3, 4). Graph the circle and give the domain and range of the relation.
Solution Substitute h = –3, k = 4, and r = 6 into the equation of a circle.
22
222
)4()3(36
)4())3((6
yx
yx
Copyright © 2007 Pearson Education, Inc. Slide 6-6
6.1 Graphing Circles with the Graphing Calculator
Example Use the graphing calculator to graph the circle in a square viewing window.
Solution
922 yx
.9 and 9Let
9
9
9
22
21
2
22
22
xyxy
xy
xy
yx
Copyright © 2007 Pearson Education, Inc. Slide 6-7
6.1 Graphing Circles with the Graphing Calculator
• TECHNOLOGY NOTES:– Graphs in a nondecimal window may not be
connected
– Graphs in a rectangular (non-square) window look like an ellipse
Copyright © 2007 Pearson Education, Inc. Slide 6-8
6.1 Finding the Center and Radius of a Circle
Example Find the center and radius of the circle with equation
Solution Our goal is to obtain an equivalent equation of the formWe complete the square in both x and y.
2 26 10 25 0.x x y y
.)()( 222 kyhxr
2 2
2 2
2 2
2 2 2
6 10 25
( 6 9) ( 10 25) 25 9 25
( 3) ( 5) 9
( 3) ( 2) 3
x x y y
x x y y
x y
x y
The circle has center (3, –2) with radius 3.
Copyright © 2007 Pearson Education, Inc. Slide 6-9
6.1 Equations and Graphs of Parabolas
• For example, let the directrix be the line y = –c and the focus be the point F with coordinates (0, c).
A parabola is a set of points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line, the directrix.
Copyright © 2007 Pearson Education, Inc. Slide 6-10
6.1 Equations and Graphs of Parabolas
• To get the equation of the set of points that are the same distance from the line y = –c and the point (0, c), choose a point P(x, y) on the parabola. The distance from the focus, F, to P, and the point on the directrix, D, to P, must have the same length.
cyxcycycycyx
cycycycyx
cyxxcyx
DPdFPd
422
22
))(()()()0(
),(),(
2
22222
22222
2222
Copyright © 2007 Pearson Education, Inc. Slide 6-11
6.1 Parabola with a Vertical Axis
• The focal chord through the focus and perpendicular to the axis of symmetry of a parabola has length |4c|.
– Let y = c and solve for x.
The endpoints of the chord are ( x, c), so the length is |4c|.
The parabola with focus (0, c) and directrix y = –c has equation x2 = 4cy. The parabola has vertical axis x = 0, opens upward if c > 0, and opens downward if c < 0.
c cxcx
cyx
2or 24
422
2
Copyright © 2007 Pearson Education, Inc. Slide 6-12
6.1 Parabola with a Horizontal Axis
• Note: a parabola with a horizontal axis is not a function.
• The graph can be obtained using a graphing calculator by solving y2 = 4cx for y:
Let and graph each half of the parabola.
The parabola with focus (c, 0) and directrix x = –c has equation y2 = 4cx. The parabola has horizontal axis y = 0, opens to the right if c > 0, and to the left if c < 0.
.2 cxy cxycxy 2 and 2 21
Copyright © 2007 Pearson Education, Inc. Slide 6-13
6.1 Determining Information about Parabolas from Equations
Example Find the focus, directrix, vertex, and axis
of each parabola.(a)
Solution(a)
xyyx 28 (b)8 22
284
cc
Since the x-term is squared, the parabola is vertical, with focus at (0, c) = (0, 2) and directrix y = –2. The vertex is (0, 0), and the axis is the y-axis.
Copyright © 2007 Pearson Education, Inc. Slide 6-14
6.1 Determining Information about Parabolas from Equations
(b)
The parabola is horizontal, with focus (–7, 0), directrix x = 7, vertex (0, 0), and x-axis as axis of the parabola. Since c is negative, the graph opens to the left.
7284
cc
Copyright © 2007 Pearson Education, Inc. Slide 6-15
6.1 Writing Equations of Parabolas
Example Write an equation for the parabola with vertex (1, 3) and focus (–1, 3).
Solution Focus lies left of the vertex implies theparabola has
- a horizontal axis, and- opens to the left.
Distance between vertex and focus is 1–(–1) = 2, so c = –2.
)1(8)3()1)(2(4)3(
2
2
xyxy
Copyright © 2007 Pearson Education, Inc. Slide 6-16
6.1 An Application of Parabolas
Example Signals coming in parallel to the axis of a parabolic reflector are reflected to the focus, thus concentrating the signal. The Parkes radio telescope has a parabolic dish shape with diameter 210 feet and depth 32
feet.
Copyright © 2007 Pearson Education, Inc. Slide 6-17
6.1 An Application of Parabolas
(a) Determine the equation describing the cross section.(b) The receiver must be placed at the focus of the parabola.
How far from the vertex of the parabolic dish should the receiver be placed?
Solution(a) The parabola will have the form y = ax2 (vertex at the
origin) and pass through the point ).32,105(32,2210
.025,11
32
by described becan section cross theso,025,11
3210532
)105(32
2
2
2
xy
a
a
Copyright © 2007 Pearson Education, Inc. Slide 6-18
6.1 An Application of Parabolas
(b) Since
The receiver should be placed at (0, 86.1), or 86.1 feet above the vertex.
,025,11
32 2xy
.1.86128
025,1132025,11
4
14
c
c
ac