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WELCOME TO OUR CLASS – 10D
Teacher: Dương Đức Thìn
Ninh Binh Department of Education and Training
NHO QUAN C HIGH SCHOOL
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Teacher: Dương Đức Thìn
Some pictures of the circle
Clock
Car wheel
Coins
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Circle equation with given center and radius
1
Comment
2
Practice
The question of a fantgent to a circle (lesson 2)
3
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Teacher: Dương Đức Thìn
Teacher: Dương Đức Thìn
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
To specify the circle,
which elements we need know?
It is center and radius.
Teacher: Dương Đức Thìn
In the Oxy plane, given the circle (C) with :
+ Center (a;b)
+ Radius R
+ M(x,y) (C)
 M = R
if
1. Circle equation with given center and radius
R
x
o

b
a
y
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
which condition is M satified?
Teacher: Dương Đức Thìn
 (x – a)2 + (y - b)2 = R2
In the Oxy plane, given the circle (C) with :
+ Center (a;b)
+ Radius R
+ M(x,y) (C)
 M = R
The equation (x – a)2 + (y - b)2 = R2 (1) is called the equation of the circle with center (a,b), radius R
1. Circle equation with given center and radius
R
x
o

b
a
y
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
Example 1. In the Oxy plane, the equation of the circle
with the center I(2;-3), radius R = 4 is:
A.
B.
C.
D.
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
Example 2 . In the Oxy plane, given the circle (C):
with center, radius is:
A. Center I(3; -4), Radius R= 25
B. Center I(-3; 4), Radius R= 25
C. Center I(3; -4), Radius R= 5
D. Center I(-3; 4), Radius R= 5
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
Example 3. Write the equation of the circle with the center I(1;4) and passes through B(2;6) >>> Group 1, 2
Example 4. Write the equation of the circle has diameter AB with A(3;-2) and B(1;4). >>>Group 3, 4

1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
Example 3. Write the equation of the circle with the center I(1;4) and passes through B(2;6)
+ The radius of the circle is:
+ The equation of the circle is:
B

Solve
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
+ The center of the circle is midpoint of AB
+ The radius of the circle is:
+ The equation of the circle is:
B

A
Example 4. Write the equation of the circle has diameter AB with A(3;-2) and B(1;4).
Solve
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
Example 5.
Note:
The circle equation with center O(0;0) and radius R is:
x2 + y2 = R2
1. Circle equation with given center and radius
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
+ The circle equation
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Comments
2
Develop (1) converter into the quadratic equation with x2 and y2
Teacher: Dương Đức Thìn
+ The circle equation
Can be rewritten in the form
where
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Comments
2
Conditions exists of the equation
x2 + y2 – 2ax – 2by + c = 0
Teacher: Dương Đức Thìn
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Comments
2
1. The coefficients of x2 and y2 are same.
2. a2 + b2 – c > 0.
- The coefficients of x2 and y2 are same.
Teacher: Dương Đức Thìn
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Comments
2
Teacher: Dương Đức Thìn
 
Example 1. In the following equations, the circle equation is:
A.
B.
C.
D.
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Comments
2
Teacher: Dương Đức Thìn
 
Example 2 . In the Oxy plane, given the circle (C):
with center and radius is:
A. Center I(-1; 2), Radius R=9
B. Center I(1; -2), Radius R=3
C. Center I(2; -4), Radius R=9
D. Center I(-2; 4), Radius R= 3
Comments
2
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
 
Example 3. State which of the following is the circle equation?
Find the centers and radius (if yes).
a) 2x2 + y2 – 8x + 2y – 1 = 0. Group 1,2
b) x2 + y2 + 2x – 4y – 4 = 0. Group 1,2
c) x2 + y2 – 2x – 6y + 20 = 0 Group 3,4
d) x2 + y2 + 6x + 2y + 9 = 0. Group 3,4
Comments
2
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
 
Example 3. State which of the following is the circle equation?
Find the centers and radius (if yes).
a) 2x2 + y2 – 8x + 2y – 1 = 0.
b) x2 + y2 + 2x – 4y – 4 = 0.
c) x2 + y2 – 2x – 6y + 20 = 0
d) x2 + y2 + 6x + 2y + 9 = 0
No, because the coefficients
of x2 and y2 are different
No, because a2 + b2 – c < 0.
Center I(-1;2), radius R = 3
Center K(-3;-1), radius R = 1
Comments
2
UNIT 2. CIRCLE EQUATIONS (LESSON 1)
Teacher: Dương Đức Thìn
SUMMARY
1. Circle equation with given center and radius
2. Identification the circle equation
If , then the equation
(2) is the circle equation
with center , radius
with center , radius R

Trong khoa học kỹ thuật người ta vận dụng phương trình đường tròn để đưa ra quỹ đạo bay của một vệ tinh bay rất gần trái đất, cách tâm trái đất một khoảng không đổi.
Toán học và thực tiễn
Teacher: Dương Đức Thìn
Suppose: Earth`s artificial satellites fly very close to the earth in a circle orbit, about 1,000 km (Earth`s radius about 6400 km). Know that at some point the center of the earth is at coordinates I(1; 2). Write equation orbit of satellites?
Answer: (x-1)2 + (y-2)2 = 74002
The real problem
Teacher: Dương Đức Thìn
Suppose: Earth`s artificial satellites fly very close to the earth in a circle orbit, about 1,000 km (Earth`s radius about 6400 km). Know that at some point the center of the earth is at coordinates I(1; 2). Write equation orbit of satellites?
The circle equation: (x-1)2 + (y-2)2 = 74002
The real problem
Teacher: Dương Đức Thìn
Radius R = 6400 + 1000 = 7400
Giả sử: Vệ tinh nhân tạo của trái đất bay rất gần với trái đất theo quỹ đạo đường tròn, cách trái đất một khoảng 1000km (bán kính trái đất khoảng 6400Km). Biết rằng tại một thời điểm nào đó tâm của trái đất ở tọa độ I(1; 2). Viết phương trình quỹ đạo của vệ tinh?
Đáp án: (x-1)2 + (y-2)2 = 74002
Bài toán thực tế
Teacher: Dương Đức Thìn