circles and ellipses

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Circles and Ellipses Lesson 10.3-10.4

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Page 1: Circles and ellipses

Circles and EllipsesLesson 10.3-10.4

Page 2: Circles and ellipses

Slicing these cones with a plane at different angles produces different conic sections.

Page 3: Circles and ellipses

For example, you can describe a circle as a locus of points that are a fixed distance from a fixed point.

Definition of a Circle◦ A circle is a locus of points P in a plane, that are a

constant distance, r, from a fixed point, C. Symbolically, PC r. The fixed point is called the center and the constant distance is called the radius.

Page 4: Circles and ellipses
Page 5: Circles and ellipses

EX 1 Write an equation of a circle with center (3, -2) and a radius of 4.

2 2 2+x h y k r

22 23 + 2 4x y

2 23 + 2 16x y

Page 6: Circles and ellipses

EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10.

2 2 2+x h y k r

2 2 24 + 0 5x y

2 24 +y 25x

Page 7: Circles and ellipses

EX 4 Find the coordinates of the center and the measure of the radius.

2 2 26 + 3 25x y

Page 8: Circles and ellipses

Converting from Graphing Form to Standard

1. Move the x terms together and the y terms together.

2. Move C to the other side.3. Complete the square (as

needed) for x.4. Complete the square(as

needed) for y.5. Factor the left & simplify the

right.

Page 9: Circles and ellipses

2 24 6 3 x x y y

9. Write the standard equation of the circle. State the center & radius.

2 2 4 6 3 0 x y x y

Center: (-2, 3) radius: 4

2 22 3 16 x y

2 24 6 9394 4 x x y y

Page 10: Circles and ellipses

Find the equation of the circle whose endpoints of a diameterare (11, 18) and (-13, -20):

Center is the midpoint of the diameter

11 13 18 20, 1 1,

2 2

Radius uses distance formula

2 2

1 2 1 2r x x y y

2 2r 11 1 18 1

r 505

2 2r 13 1 20 1

r 505

22 21 1 05x 5y

Page 11: Circles and ellipses
Page 12: Circles and ellipses

Write an equation in standard form of an ellipse that has a vertex at

(0, –4), a co-vertex at (3, 0), and is centered at the origin.

Writing an Equation of an Ellipse

Since (0, –4) is a vertex of the ellipse, the other vertex is at (0, 4), and the major axis is vertical.Since (3, 0) is a co-vertex, the other co-vertex is at (–3, 0), and the minor axis is horizontal.So, a = 4, b = 3, a2 = 16, and b2 = 9.

+ = 1 Standard form for an equation of an ellipse with a vertical major axis.

(x-h) 2

b2(y-k) 2

a2

+ = 1 Substitute 9 for b2 and 16 for a2.(x-0) 2

9 (y-0) 2

16

An equation of the ellipse is + = 1.x 2

9 y 2

16

Page 13: Circles and ellipses

Graph and Label b) Find coordinates of vertices,

covertices, foci

Center = (-3,2) Horizontal ellipse since the a²

value is under x terms Since a = 3 and b = 2 Vertices are 3 points left and

right from center (-3 ± 3, 2) Covertices are 2 points up and

down (-3, 2 ± 2) Now to find focus points Use c² = a² - b² So c² = 9 – 4 = 5 c² = 5 and c = ±√5 Focus points are √5 left and

right from the center F(-3 ±√5 , 2)

14

)2y(

9

)3x( 22

• a) GRAPH• Plot Center (-3,2)• a = 3 (go left and

right)• b = 2 (go up and

down)

Page 14: Circles and ellipses

Find the foci of the ellipse with the equation 9x2 + y2 = 36. Graph the ellipse.

Working Backwards

9x2 + y2 = 36

Since 36 > 4 and 36 is with y2, the major axis is vertical, a2 = 36, and b2 = 4.

+ = 1 Write in standard form.x 2

4y 2

36

c2 = a2 – b2 Find c.

= 36 – 4 Substitute 4 for a2 and 36 for b2.

= 32

The major axis is vertical, so the coordinates of the foci are (0, ±c). The foci are: (0, 4 2 ) and (0, – 4 2).

c = ± 32 = ± 4 2