Write an equation of a circle with diameter
A
B
¯
.
eh b bar , .
-
A
(
0
,
0
)
,
B
(
8
,
6
)
eh open 0 comma 0 close comma b open 8 comma 6 close
-
A
(
3
,
0
)
,
B
(
7
,
6
)
eh open 3 comma 0 close comma b open 7 comma 6 close
-
A
(
1
,
1
)
,
B
(
5
,
5
)
eh open 1 comma 1 close comma b open 5 comma 5 close
-
Reasoning Describe the graph of
x
2
+
y
2
=
r
2
x squared , plus , y squared , equals , r squared when
r
=
0
.
r equals 0 .
Determine whether each equation is the equation of a circle. Justify your answer.
-
(
x
−
1
)
2
+
(
y
+
2
)
2
=
9
open x minus . 1 , close squared , plus , open y plus . 2 , close squared . equals 9
-
x
+
y
=
9
x plus . y equals 9
-
x
+
(
y
−
3
)
2
=
9
x plus . open y minus . 3 , close squared . equals 9
-
Think About a Plan Find the circumference and area of the circle whose equation is
(
x
−
9
)
2
+
(
y
−
3
)
2
=
64
.
open x minus . 9 , close squared , plus , open y minus . 3 , close squared . equals 64 . Leave your answers in terms of
π
.
pi .
- What essential information do you need?
- What formulas will you use?
- Write an equation of a circle with area
36
π
36 pi and center
(
4
,
7
)
.
open 4 comma 7 close .
- What are the x- and y-intercepts of the line tangent to the circle
(
x
−
2
)
2
+
(
y
−
2
)
2
=
5
2
open x minus . 2 , close squared , plus , open y minus . 2 , close squared , equals , 5 squared at the point (5, 6)?
- For
(
x
−
h
)
2
+
(
y
−
k
)
2
=
r
2
,
open x minus . h , close squared , plus , open y minus . k , close squared , equals , r squared , comma show that
y
=
r
2
−
(
x
−
h
)
2
+
k
y equals . square root of r squared , minus . open , x minus h , close squared end root . plus k or
y
=
−
r
2
−
(
x
−
h
)
2
+
k
.
y equals negative . square root of r squared , minus . open , x minus h , close squared end root . plus k .
Sketch the graphs of each equation. Find all points of intersection of each pair of graphs.
-
x
2
+
y
2
=
13
y
=
−
x
+
5
table with 2 rows and 1 column , row1 column 1 , x squared , plus , y squared , equals 13 , row2 column 1 , y equals negative x plus 5 , end table
-
x
2
+
y
2
=
17
y
=
−
1
4
x
table with 2 rows and 1 column , row1 column 1 , x squared , plus , y squared , equals 17 , row2 column 1 , y equals negative , 1 fourth , x , end table
-
x
2
+
y
2
=
8
y
=
2
table with 2 rows and 1 column , row1 column 1 , x squared , plus , y squared , equals 8 , row2 column 1 , y equals 2 , end table
-
x
2
+
y
2
=
20
y
=
−
1
2
x
+
5
table with 2 rows and 1 column , row1 column 1 , x squared , plus , y squared , equals 20 , row2 column 1 , y equals negative , 1 half , x plus 5 , end table
-
(
x
+
1
)
2
+
(
y
−
1
)
2
=
18
y
=
x
+
8
table with 2 rows and 1 column , row1 column 1 , open x plus 1 , close squared , plus open y minus 1 , close squared , equals 18 , row2 column 1 , y equals x plus 8 , end table
-
(
x
−
2
)
2
+
(
y
−
2
)
2
=
10
y
=
−
1
3
x
+
6
table with 2 rows and 1 column , row1 column 1 , open x minus 2 , close squared , plus open y minus 2 , close squared , equals 10 , row2 column 1 , y equals negative , 1 third , x plus 6 , end table
Graphing Calculator Use a graphing calculator to convince yourself that the given line is not tangent to the circle
x
2
+
y
2
=
25
.
x squared , plus , y squared . equals 25 . Explain what you did.
-
y
=
−
5
x
+
26
y equals , minus . 5 x plus 26
-
3
x
+
5
y
=
29
3 x plus . 5 y equals 29
-
Writing Why it is not possible to conclude that a line and a circle are tangent by viewing their graphs?
C Challenge
-
Geometry in 3 Dimensions The equation of a sphere is similar to the equation of a circle. The equation of a sphere with center (h, j, k) and radius r is
(
x
−
h
)
2
+
(
y
−
j
)
2
+
(
z
−
k
)
2
=
r
2
.
M
(
−
1
,
3
,
2
)
open x minus . h , close squared , plus , open y minus . j , close squared , plus , open z minus . k , close squared , equals , r squared , . . m open negative 1 comma 3 comma 2 close is the center of a sphere passing through T(0, 5, 1). What is the radius of the sphere? What is the equation of the sphere?
- The concentric circles
(
x
−
3
)
2
+
(
y
−
5
)
2
=
64
open x minus . 3 , close squared , plus , open y minus . 5 , close squared . equals 64 and
(
x
−
3
)
2
+
(
y
−
5
)
2
=
25
open x minus . 3 , close squared , plus , open y minus . 5 , close squared . equals 25 form a ring. The lines
y
=
2
3
x
+
3
y equals , 2 thirds . x plus 3 and
y
=
5
y equals 5 intersect the ring, making four sections. Find the area of each section. Round your answers to the nearest tenth of a square unit.
