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Conic Sections

Date post: 03-Jan-2016
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Conic Sections. An Introduction. Conic Sections - Introduction. A conic is a shape generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant. Conic Sections - Introduction. - PowerPoint PPT Presentation
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An Introduction
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Page 1: Conic Sections

An Introduction

Page 2: Conic Sections

Conic Sections - IntroductionA conic is a shape

generated by intersecting two lines at a point and rotating one line around the other while keeping the angle between the lines constant.

Page 3: Conic Sections

Conic Sections - IntroductionThe resulting

collection of points is called a right circular cone. The two parts of the cone intersecting at the vertex are called nappes.

Vertex

Nappe

Page 4: Conic Sections

Conic Sections - IntroductionA “conic” or conic

section is the intersection of a plane with the cone.

The plane can intersect the cone at the vertex resulting in a point.

Page 5: Conic Sections

Conic Sections - IntroductionThe plane can

intersect the cone perpendicular to the axis resulting in a circle.

Page 6: Conic Sections

Conic Sections - IntroductionThe plane can

intersect one nappe of the cone at an angle to the axis resulting in an ellipse.

Page 7: Conic Sections

Conic Sections - IntroductionThe plane can

intersect one nappe of the cone at an angle to the axis resulting in a parabola.

Page 8: Conic Sections

Conic Sections - IntroductionThe plane can intersect

two nappes of the cone resulting in a hyperbola.

Page 9: Conic Sections

Graph each equation. Describe the graph, find the lines of symmetry, x and y intercepts, domain, and range.

1.

2.

3.

1622 yx

3649 22 yx

422 yx

Page 10: Conic Sections

162.3

81.2

100425.1

22

22

22

yx

yx

yx

Page 11: Conic Sections

What is a circle?A circle is the set

of points equally distant from one central point.

The central point is called the center.

••

Center

Page 12: Conic Sections

What does r represent?The distance from the center to the curve of

the circle is called the radius.

r

Page 13: Conic Sections

What does d represent?The diameter is the distance across the

circle.

d

Page 14: Conic Sections

Assume that (x,y) are the coordinates of a point on the circle.

Use the distance formula to find the radius.

(x,y)

(h,k)

Page 15: Conic Sections

Equation of a circler2=(x - h)2 + (y – k)2

Let’s investigate!

Page 16: Conic Sections

Example #1Find the equation of a circle whose center is

at (2, -4) and the radius is 5.

Let’s check our answer.

Page 17: Conic Sections

Example #2Write an equation of a circle if the endpoints

of a diameter are at (5,4) and (-2, -6).Hint: Draw a picture, then find the

center and radius.

Page 18: Conic Sections

Example #3Find the center and radius of the circle with equation x2 + y2 = 25.

Graph the circle.

Page 19: Conic Sections

Example #4Find the center and radius of the circle with equation

x2 + y2 – 4x + 8y – 5 = 0.

Page 20: Conic Sections

On your own…

1. Find center and radius and graph: x2 + y 2 -10x +8y = -40

2. Write an equation for a circle that passes through the point (-1, 4) with a center at (-3, 6).

Page 21: Conic Sections

Assignment:10.1 p. 550 #1, 3, 5, 17-28, 62

10.3 p. 564 #27-31 odd, 35, 43, 45, 47, 49, 61, 63, 73, (78 graph)


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