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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 ) :