Chuong III 4 Duong tron

Example the equation ofcircle the circle center Example11: :Write The equation of the withwith center I2; -I2; 3 - 3 2 2 and andradius radiusRR==55... Write the equation of the circle C:[r]

Check up

1 Write the formula of calculated distance

between two points I(a;b) and M(x;y)?

  2  2

IM  x a  y b

2

3

(C)

x

y

O

5

B

D

A IA 2 10  R  A( ).C

2 Oxy, given the circle (C) with center I(2;3) and radius R=5 A(-4;5), B(-2;0), D(3;2)? Find point on the circle (C)?

IB   R B  C

M(x;y)

x

y

O R M(x; y) (C) IM = R

1 Circle equations with given center and radius.

(1) is called the equation of the circle

with center I(a;b) and radius R

Example 1: The equation of the circle with center I(2; - 3)

and radius R = 5 is:

Note: The equation of the circle with the origin O as its

center and radius R is .

 x a2  y b2 R2  1



x  y  R

 x  22  y  32  25

§2 CIRCLE EQUATIONS

State condition for the point M(x;y) is

on the circle (C)?

Example 1: Write the equation of the circle with center I(2; - 3) and radius R = 5

y

x

O

A(3;-4)

Example 2: Given two points A(3;-4), B(-3;4)

Write the equation of the circle (C):

a) With center A and pass through B

b) With diameter AB

A Equation

1 (x + 2) 2 +(y - 6) 2 = 1

2 (x - 1) 2 + y 2 = 25

3 x 2 + y 2 = 3/2

4 4x 2 + (2y + 6) 2 = 16

Example 3:

Match each equation in A with a suitable circle in B

B Circle

a Center (0; -3), radius R = 2

b Center (0; 0), radius

c Center (-2; 6) , radius R = 1

d Center (0; -6), radius R = 4

e Center(1; 0), radius R = 5

6 2

R 

Conclusion: (2) is the equation of the circle (C) if and only if a2 + b2 – c > 0 Then the circle has center I(a; b) and radius

Develop the equation (1)?

 x a 2   y b 2 R2(1)

(2)  x a    y b    a b c

2 2

R  a  b  c

2 Comments

When is (2) equation of the circle?

Example 4:Given the equation:

Is this a circle equation?If it is, find the center and radius of the circle?

Game: “Lucky Star”

will take turns to choose a star you like and answer the questions inside it

If your answer is correct, you will get 10 points, if not the chance will be given to other groups

However, if you choose a lucky star, you will get

10 points without having to answer any question

6

5

4

3

Lucky Star

2

Given the equation:

Choose the correct answer among A, B, C or D.

A (1) isn’t equation of circle

B The circle (1) has center I(4; -1).

C The circle (1) has C The circle (1) has radius R = 3.

D The circle (1) has center I(-4; 1).

Question 1

2 2

2 – 8 2 1 0(1)x  y x  y  

Question 3

Which of the following is the circle equation?

(A)

(B)

(C)

(D)

2 2 – 4 – 8 +1 0 2

2 – 4 2 6y + 20 0

2 2

4 – 10 6 2 0x  y x  y  

2 – 4 6 12 0 2

Congratulations! You’ve just got a lucky star!

Given the circle (C):

Find the wrong proposition in the following propositions:

(A) The circle (C) has center I(1; 2);

(B) The circle (C) has radius R = 5;

(C) The circle (C) passes through the point M(2; 2);

(D) The circle (C) does not pass through the point A(1; 1).

2 2 + 4 – 20 02

The circle (C):

with center I and radius R is:

Question 2

( ) ( 1;1), R 1;

A I I

C I

Congratulations! You’ve just got a lucky star!

10 points

 x a 2  y b 2 R2 (1)

2 2

R  a b  c

1 The equation of the circle with center I(a;b) and radius R isThe equation of the circle with center I(a;b) and radius R

2 The equation:

is the equation of the circle (C) if and only if

Then the circle has center I(a; b) and radius

Homework:

- Do exercises 1 – 5 / page 83, 84

2 2 0.

a b  c 