Circles and ellipses
Transcript of Circles and ellipses
Circles and EllipsesLesson 10.3-10.4
Slicing these cones with a plane at different angles produces different conic sections.
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.
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
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
EX 4 Find the coordinates of the center and the measure of the radius.
2 2 26 + 3 25x y
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.
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
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
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
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)
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