Upcoming SlideShare
×

# Circles and ellipses

603 views

Published on

Published in: Spiritual, Technology
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
603
On SlideShare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
7
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Circles and ellipses

1. 1. Lesson 10.3-10.4
2. 2.  Slicing these cones with a plane at different angles produces different conic sections.
3. 3.  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.
4. 4. 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 2 3 + 2 4x y 2 2 3 + 2 16x y
5. 5. 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 2 4 + 0 5x y 2 2 4 +y 25x
6. 6. EX 4 Find the coordinates of the center and the measure of the radius. 2 2 2 6 + 3 25x y
7. 7. 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.
8. 8. 2 2 4 6 3x x y y 2 2 4 6 3 0x y x y Center: (-2, 3) radius: 4 2 2 2 3 16x y 2 2 4 6 9394 4x x y y
9. 9. Find the equation of the circle whose endpoints of a diameter are (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 2 r 11 1 18 1 r 505 2 2 r 13 1 20 1 r 505 22 2 1 1 05x 5y
10. 10. 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. 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
11. 11.  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) 1 4 )2y( 9 )3x( 22 • a) GRAPH • Plot Center (-3,2) • a = 3 (go left and right) • b = 2 (go up and down)
12. 12. Find the foci of the ellipse with the equation 9x2 + y2 = 36. Graph the ellipse. 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 4 y 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