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# Ellipses and Circles - PowerPoint PPT Presentation

Ellipses and Circles. Section 10.3. 1 st Definition of a Circle. A circle is a conic section formed by a plane intersecting one cone perpendicular to the axis of the double-napped cone. The degenerate conic section that is associated with a circle is a point. 2 nd Definition of a Circle.

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## PowerPoint Slideshow about ' Ellipses and Circles' - clancy

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### Ellipses and Circles

Section 10.3

1st Definition of a Circle

2 one cone perpendicular to the axis of the double-napped cone.nd Definition of a Circle

Example 1 same distance from a

• Express in standard form the equation of the circle centered at (-2, 3) with radius 5.

Example 2 same distance from a

• Express in standard form the equation of the circle with center at the origin and radius of 4. Sketch the graph.

Example 3 same distance from a

• Find the center and radius of the circle with the equation

• Center:

Example 4 same distance from a

• Write the equation for each circle described below.

• a. The circle has its center at (8, -9) and passes through the point at (4, -6).

1 same distance from a st Definition of an Ellipse

2 one cone not perpendicular to the axis of the double-napped cone.nd Definition

• An ellipse is the set of all points ( one cone not perpendicular to the axis of the double-napped cone.x, y) in a plane,

• the sum of whose distances from two distinct

• fixed points (foci) is constant.

• d1 + d2 = constant

d1

d2

• The line through the foci intersects the ellipse one cone not perpendicular to the axis of the double-napped cone.

• 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.

minor axis

major axis

center

vertex

vertex

General Equation of an Ellipse one cone not perpendicular to the axis of the double-napped cone.

• Ax2 + Cy2 + Dx + Ey + F = 0

• If A = C, then the ellipse is a circle.

Standard Equation of an one cone not perpendicular to the axis of the double-napped cone.Ellipse

where the major axis is horizontal.

where the major axis is vertical.

• Find the center, vertices, the endpoints of the minor axis, foci, eccentricity, and graph for the ellipses given in standard form.

• a =

• b =

• c =

• center: (

• vertices:

• endpoints of the minor axis:

• foci:

• eccentricity:

F (2

F1

V1

C

V2

• For the following ellipse, find the center, a, b, c, vertices, the endpoints of the minor axis, foci, eccentricity, and graph.

• 16x2 + y2 − 64x + 2y + 49 = 0

• What must you do to the general equation above to do this example?

• Complete the square twice.

• 16 (x2 + y2 − 64x + 2y + 49 = 0

• 16x2 − 64x+ y2+ 2y= −49

• center: (

• vertices:

• endpoints of the minor axis:

• foci:

• eccentricity:

1

a =

4

b =

What type of ellipse is this ellipse?

vertical ellipse?

(2, −1)

(2, 3), (2, −5)

(3, −1), (1, −1)

V (1

F1

C

F2

V2

• Write the equation of each ellipse in standard form.

• A. Endpoints of the major axis are at (0, ±10) and whose foci are at (0, ±8).

• center: (0, 0)

• vertical ellipse

• a = 10; c = 8

• b =

• B. (The endpoints of the major axis are at (10, 2) and (–8, 2). The foci are at (6, 2) and (–4, 2).

• C. (The major axis is 20 units in length and parallel to the y-axis. The minor axis is 6 units in length. The center is located at

• (4, 2).