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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

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

- A circle is the set of all points P in a plane that are the same distance from a given point.
- The given distance is the radius of the circle, and the given point is the center of the circle.
- Standard form of a circle with center C (h, k) and radius r is

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:
- Radius =

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).

- b. The endpoints of a diameter are at (1, 8) and (1, -4). same distance from a

End of 1st Day

1 same distance from a st Definition of an Ellipse

- An ellipse is a conic section formed by a plane intersecting one cone not perpendicular to the axis of the double-napped cone.
- The degenerate conic section that is associated with an ellipse is also a point.

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

- The standard form of the equation of an ellipse, with center (h, k) and major and minor axes of lengths 2a and 2b respectively, where 0 < b < a,

where the major axis is horizontal.

where the major axis is vertical.

- The foci lie on the major axis, (c units from the center, with c2 = a2 – b2.
- The eccentricity of an ellipse is

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:

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)

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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