Distance and Circles

( h , k )

r

222 kyhxr

Standard form for the equation of a circle :

- center ( h , k )- radius ( r )

Distance and Circles

( h , k )

r

222 kyhxr

Standard form for the equation of a circle :

- center ( h , k )- radius ( r )

The distance from the center of the circle to any point ( x , y ) ON the circle is the RADIUS

Distance and Circles

( h , k )

r

222 kyhxr

Standard form for the equation of a circle :

- center ( h , k )- radius ( r )

When the equation of the circle is given in the form;

022 dcybxayax

You must rewrite the equation in standard form by completing the square…

Distance and Circles

( h , k )

r

222 kyhxr

Standard form for the equation of a circle :

- center ( h , k )- radius ( r )

When the equation of the circle is given in the form;

022 dcybxayax

You must rewrite the equation in standard form by completing the square…

Let’s look at the standard form first…

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

To get ( x – 4 ), h would have to be +4

- ( x – h )2 = ( x – (+4))2 = (x – 4 )2

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

To get ( x – 4 ), h would have to be +4

- ( x – h )2 = ( x – (+4))2 = (x – 4 )2

To get ( y + 3 ), k would have to be - 3

- ( y – k )2 = ( y – ( -3))2 = ( y + 3 )2

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

To get ( x – 4 ), h would have to be +4

- ( x – h )2 = ( x – (+4))2 = (x – 4 )2

To get ( y + 3 ), k would have to be - 3

- ( y – k )2 = ( y – ( -3))2 = ( y + 3 )2

CENTER = ( 4 , - 3 )

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

To get ( x – 4 ), h would have to be +4

- ( x – h )2 = ( x – (+4))2 = (x – 4 )2

To get ( y + 3 ), k would have to be - 3

- ( y – k )2 = ( y – ( -3))2 = ( y + 3 )2

There is a short cut…just use the

OPPOSITE sign you see in front

of h and k

CENTER = ( 4 , - 3 )

Distance and Circles

( h , k )

r

22 3436 yx

Find the center and radius of the circle whose equations is :

To get ( x – 4 ), h would have to be +4

- ( x – h )2 = ( x – (+4))2 = (x – 4 )2

To get ( y + 3 ), k would have to be - 3

- ( y – k )2 = ( y – ( -3))2 = ( y + 3 )2

There is a short cut…just use the

OPPOSITE sign you see in front

of h and k

CENTER = ( 4 , - 3 ) and if r2 = 36, r = 6

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___10____16 22 yyxx

Rewrite the equation getting your x’s and y’s together.

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___10____16 22 yyxx

Rewrite the equation getting your x’s and y’s together.

Move any integer to the other side of the equation.

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___10____16 22 yyxx

Rewrite the equation getting your x’s and y’s together.

Move any integer to the other side of the equation.

Leave one blank space behind each x/y group and 2 behind your #

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___10____16 22 yyxx

Write the standard equation form leaving blanks in the spots in squares… also leave a few lines space for the next step in between…

______ 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___2

10____

2

16 22 yyxx

To complete the square, divide the linear x and y coefficient by 2…

______ 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___2

10____

2

16 22 yyxx

To complete the square, divide the linear x and y coefficient by 2…the answer will fill in the blank spaces in the standard form…

__58 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

___ ___80___2

10____

2

16 22 yyxx

__58 22 yx

Next, square those answers and fill in the blank spaces on both sides of the equation…

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Next, square those answers and fill in the blank spaces on both sides of the equation…

__58 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Then, complete the addition on the right side and fill in the the last blank in the standard form…

__58 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Then, complete the addition on the right side and fill in the the last blank in the standard form…

958 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Let’s clean up our double signs….

958 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Center = ( - 8 , - 5 )

r = 3

958 22 yx

Distance and Circles

Completing the square – forcing an expression into a perfect square

trinomial

EXAMPLE : Find the center and radius of a circle defined by the equation :

080101622 yxyx

256480252

1064

2

16 22 yyxx

Center = ( - 8 , - 5 )

r = 3

958 22 yx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

05

10

5

30

5

20

5

5

5

5 22

yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

05

10

5

30

5

20

5

5

5

5 22

yxyx

026422 yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

______2____6___4 22 yyxx

__________ 22 yx

Now complete your square…

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

______2____2

6___

2

4 22 yyxx

____32 22 yx

Now complete your square…

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

94292

64

2

4 22 yyxx

____32 22 yx

Fill in the squares…

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

94292

64

2

4 22 yyxx

1532 22 yx

EXAMPLE #2 : Find the center and radius of a circle defined by the equation :

010302055 22 yxyx

026422 yxyx

94292

64

2

4 22 yyxx

1532 22 yx

EXAMPLE #3 : Find the center and radius of a circle defined by the equation :

047121822 yxyx

EXAMPLE #3 : Find the center and radius of a circle defined by the equation :

047121822 yxyx

EXAMPLE #3 : Find the center and radius of a circle defined by the equation :

047121822 yxyx

EXAMPLE #3 : Find the center and radius of a circle defined by the equation :

047121822 yxyx

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

1. Begin by substituting ( h , k ) into our circle equation :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :

EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes

thru the coordinate ( - 1 , 6 ) :