Write an Equation Using the Center and Radius
A. Write the equation of the circle with a center
at (3, –3) and a radius of 6.
(x – h)2 + (y – k)2 = r 2
(x – 3)2 + (y – (–3))2 = 62
(x – 3)2 + (y + 3)2 = 36
Answer: (x – 3)2 + (y + 3)2 = 36
Equation of circle
Substitution
Simplify.
Write an Equation Using the Center and Radius
B. Write the equation of the circle
graphed to the right.
The center is at (1, 3) and
the radius is 2.
(x – h)2 + (y – k)2 = r 2
Equation of circle
(x – 1)2 + (y – 3)2 = 22
Substitution
(x – 1)2 + (y – 3)2 = 4
Simplify.
Answer: (x – 1)2 + (y – 3)2 = 4
A. Write the equation of the circle with a center
at (2, –4) and a radius of 4.
A. (x – 2)2 + (y + 4)2 = 4
B. (x + 2)2 + (y – 4)2 = 4
C. (x – 2)2 + (y + 4)2 = 16
D. (x + 2)2 + (y – 4)2 = 16
B. Write the equation of the circle
graphed to the right.
A. x2 + (y + 3)2 = 3
B. x2 + (y – 3)2 = 3
C. x2 + (y + 3)2 = 9
D. x2 + (y – 3)2 = 9
Write an Equation Using the Center and a Point
Write the equation of the circle that has its
center at (–3, –2) and passes through (1, –2).
Step 1 Find the distance between the points to
determine the radius.
Distance Formula
(x1, y1) = (–3, –2) and
(x2, y2) = (1, –2)
Simplify.
Write an Equation Using the Center and a Point
Step 2 Write the equation using h = –3, k = –2, and
r = 4.
(x – h)2 + (y – k)2 = r 2
(x – (–3))2 + (y – (–2))2 = 42
(x + 3)2 + (y + 2)2 = 16
Answer: (x + 3)2 + (y + 2)2 = 16
Equation of circle
Substitution
Simplify.
Write the equation of the circle that has its center at
(–1, 0) and passes through (3, 0).
A. (x + 1)2 + y2 = 16
B. (x – 1)2 + y2 = 16
C. (x + 1)2 + y2 = 4
D. (x – 1)2 + y2 = 16
Graph a Circle
The equation of a circle is x2 – 4x + y2 + 6y = –9. State
the coordinates of the center and the measure of the
radius. Then graph the equation.
Write the equation in standard form by completing the
square.
x2 – 4x + y2 + 6y = –9
Original equation
x2 – 4x + 4 + y2 + 6y + 9 = –9 + 4 + 9 Complete the
squares.
(x – 2)2 + (y + 3)2 = 4
(x – 2)2 + [y – (–3)]2 = 22
Factor and simplify.
Write +3 as – (–3)
and 4 as 22.
Graph a Circle
With the equation now in standard form, you can identify
h, k, and r.
(x – 2)2 + [y – (–3)]2 = 22
(x – h)2 + [y – k]2
= r2
Answer: So, h = 2, y = –3,
and r = 2. The
center is at (2, –3),
and the radius is 2.
Which of the following is the graph of
x2 + y2 –10y = 0?
A.
B.
C.
D.
Use Three Points to Write an
Equation
Strategically located substations are extremely
important in the transmission and distribution of a
power company’s electric supply. Suppose three
substations are modeled by the points D(3, 6), E(–1,
1), and F(3, –4). Determine the location of a town
equidistant from all three substations, and write an
equation for the circle.
Understand You are given three points that lie on a
circle.
Plan
Graph ΔDEF. Construct the perpendicular
bisectors of two sides to locate the center,
which is the location of the tower. Find the
length of a radius. Use the center and
radius to write an equation.
Use Three Points to
Write an Equation
Solve
Graph ΔDEF and construct the
perpendicular bisectors of two sides.
Use Three Points to
Write an Equation
The center, C, appears to be at (4, 1). This is the
location of the tower. Find r by using the Distance
Formula with the center and any of the three points.
Write an equation.
Use Three Points to Write an
Equation
Answer: The location of a town equidistant from all
three substations is at (4,1). The equation
for the circle is (x – 4)2 + (y – 1)2 = 26.
Check
You can verify the location of the center by
finding the equations of the two bisectors
and solving a system of equations. You can
verify the radius by finding the distance
between the center and another of the three
points on the circle.
The designer of an amusement park wants to place
a food court equidistant from the roller coaster
located at (4, 1), the Ferris wheel located at (0, 1),
and the boat ride located at (4, –3). Determine the
location for the food court.
A. (3, 0)
B. (0, 0)
C. (2, –1)
D. (1, 0)
Intersections with Circles
Find the point(s) of intersection between x2 + y2 = 32
and y = x + 8.
Graph these equations on the same coordinate plane.
Intersections with Circles
There appears to be only one point of intersection.
You can estimate this point on the graph to be at
about (–4, 4). Use substitution to find the coordinates
of this point algebraically.
x2 + y2 = 32
Equation of circle.
x2 + (x + 8)2 = 32
Substitute x + 8 for y.
x2 + x2 + 16x + 64 = 32
Evaluate the square.
2x2 + 16x + 32 = 0
x2 + 8x + 16 = 0
(x + 4)2 = 0
x = –4
Simplify.
Divide each side by 2.
Factor.
Take the square root of
each side.
Intersections with Circles
Use y = x + 8 to find the corresponding y-value.
(–4) + 8 = 4
The point of intersection is (–4, 4).
Answer: (–4, 4)
Find the points of intersection between x2 + y2 = 16
and y = –x.
A. (2, –2)
B. (2, 2)
C. (–2, –2), (2, 2)
D. (–2, 2), (2, –2)