Download 10.6 Equations of Circles

Geometry
10.6 Equations of Circles
Geometry
Geometry
Objectives
• Write the equation of a circle.
• Use the equation of a circle and its
graph to solve problems.
Geometry
Finding Equations of Circles
• You can write an
equation of a circle
in a coordinate
plane if you know its
radius and the
coordinates of its
center.
6
y
(x, y)
4
(h, k)
2
x
5
-2
Geometry
Finding Equations of Circles
• Suppose the radius is r
and the center is (h, k).
• Let (x, y) be any point
on the circle.
• The distance between
(x, y) and (h, k) is r, so
• We use Distance
Formula.
• (Told you it wasn’t
going away).
6
y
(x, y)
4
(h, k)
2
x
5
-2
Finding Equations of Circles
( x  h)  ( y  k )  r
Geometry
2
2
• Square both sides
to find the standard
equation of a circle
with radius r and
center (h, k).
(x – h)2 + (y – k)2 = r2
If the center is at the
origin, then the
standard equation is
x 2 + y 2 = r 2.
6
y
(x, y)
4
(h, k)
2
x
5
-2
Geometry
Writing a Standard Equation of a
Circle
• Write the standard equation of the circle with
a center at (-4, 0) and radius 7
(x – h)2 + (y – k)2 = r2
Standard equation of a circle.
[(x – (-4)]2 + (y – 0)2 = 72
Substitute values.
(x + 4)2 + (y – 0)2 = 49
Simplify.
Geometry
Writing a Standard Equation of a
Circle
The point (1, 2) is on a circle whose center is (5, -1).
Write a standard equation of the circle.
r=
( x2  x1 ) 2  ( y2  y1 ) 2
Use the Distance Formula
r=
(5  1) 2  (1  2) 2
Substitute values.
r=
(4) 2  (3) 2
Simplify.
r=
r=
16  9
25
Simplify.
Addition Property
Square root the result.
r=5
Geometry
Ex. 2: Writing a Standard
Equation of a Circle
The point (1, 2) is on a circle whose center is (5, 1).
Write a standard equation of the circle.
(x – h)2 + (y – k)2 = r2
Standard equation of a circle.
[(x – 5)]2 + [y –(-1)]2 = 52
Substitute values.
(x - 5)2 + (y + 1)2 = 25
Simplify.
Geometry
Graphing Circles
• If you know the equation of a circle, you
can graph the circle by identifying its
center and radius.
Geometry
Graphing a circle
• The equation of a • (x+2)2 + (y-3)2 = 9
circle is
• [x – (-2)]2 + (y – 3)2=32
(x+2)2 + (y-3)2 = 9.
• The center is (-2, 3) and
Graph the circle.
the radius is 3.
First rewrite the
equation to find the
center and its
radius.
Geometry
Graphing a circle
• To graph the circle,
place the point of a
compass at (-2, 3),
set the radius at 3
units, and swing the
compass to draw a
full circle.
6
4
2
-5
-2
Geometry
Graphs of Circles
1. Rewrite the equation to find the center and radius. The
center is at (13, 4) and the radius is 4.
– (x – h)2 + (y – k)2= r2
– (x – 13)2 + (y – 4)2= 42
– (x - 13)2 + (y - 4)2 = 16
Geometry
•The center is (1, -3)
•Radius = 3
•
Geometry
•The center is (-4, -3)
•Radius = 1
•
Geometry
•(x – h)2 + (y – k)2 = r2
•(x – 8)2 + (y + 6)2 = 42
•(x – 8)2 + (y +6)2 =16
• Find Radius
3.14 = 3.14 * r2
• r=1
•(x – h)2 + (y – k)2 = r2
•(x + 12)2 + (y +11)2 = 12
•(x +12)2 + (y +11)2 =1
Geometry
• Find Radius
• Find Radius
12 * 3.14 = 2 *3.14 * r
8 * 3.14 = 2 *3.14 * r
• r = 16
• r=4
•(x – h)2 + (y – k)2 = r2
•(x – h)2 + (y – k)2 = r2
•(x + 9)2 + (y +9)2 = 62
•(x + 13)2 + (y -4)2 = 42
•(x +12)2 + (y +11)2 =36
•(x +13)2 + (y - 4)2 =16
Geometry
•The center is (4, 0)
•Radius = 3
•(x – h)2 + (y – k)2 = r2
•(x - 4)2 + (y + 0)2 = 42
•(x – 4)2 + y 2 =16
Geometry
•The center is (-3,-4)
•Radius = 2
•(x – h)2 + (y – k)2 = r2
•(x +3)2 + (y + 4)2 = 22
•(x + 3)2 + (y+4) 2 =4
Geometry
•The center is (2,4)
•Radius = √6 = 2.5
•
Geometry
•The center is (4,3)
•Radius = √2 = 1.4
•