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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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Ellipses and circles

Ellipses and Circles

Section 10.3


1 st definition of a circle
1st Definition of a Circle



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



Example 1
Example 1 same distance from a

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


Example 2
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
Example 3 same distance from a

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

  • Center:

  • Radius =


Example 4
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 st definition of an ellipse
1 same distance from a st Definition of an Ellipse



2 nd definition
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
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 ellipse
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.



Example 11
Example 1 (

  • 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


Example 21
Example 2 (

  • 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


Example 31
Example 3 (

  • 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).


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