Page 4 - For Okta_Float

Radius can also In the figure at the right, the center of the y
refer to the line circle is at the origin. By drawing a perpendicular P(x, y)
segment from the from any point P(x, y) on the circle but not on an
center to any point axis to the x-axis, you form a right triangle. The r y
on the circle. Pythagorean Theorem can be used to write an
equation that describes every point on a circle O x x
whose center is located at the origin.
2
2
x y r 2 Pythagorean Theorem
2
2
x y r 2
This is the equation for the parent graph of
all circles.
y P(x, y) Suppose the center of this circle is translated from the origin to
C(h, k). You can use the distance formula to write the equation for
r
this translated circle.
x )
2
d (x (y y ) 2 Distance formula
C(h, k) 2 1 2 1
d r, (x , y ) (x, y),
2
2
r (x h) ( 2 and (x , y ) (h, k)
2
y k)
1
1
2
2
2
2
(x h) (y k) r 2 r (x h) (y k) 2 Square each side.
O x
This equation is the standard form of the equation of a circle.
Standard The standard form of the equation of a circle with radius r and center at
Form of the (h, k) is
Equation of 2 2 2
a Circle (x h) (y k) r .

Examples 1 Write the standard form of the equation of the circle that is tangent to the
x-axis and has its center at (3, 2). Then graph the equation.
Since the circle is tangent to the x-axis, the distance
from the center to the x-axis is the radius. The center y
is 2 units below the x-axis. Therefore, the radius is 2.
2
2
(x 3) (y 2) 4
2
2
(x h) (y k) r 2 Standard form
O x
2
2
(x 3) [y (2)] 2 2 h 3, k 2, r 2
(3, 2)
2
2
(x 3) (y 2) 4
The standard form of the equation for this circle
2
2
is (x 3) (y 2) 4.
2 SEISMOLOGY Refer to the application at the beginning of the lesson.
a. Write an equation for the set of points representing all possible locations
Real World
of the earthquake’s epicenter. Let downtown Olympia, Washington, be
located at the origin.
Application
b. Graph the equation found in part a.
624 Chapter 10 Conics