![Page 1: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/1.jpg)
Distance and Circles
( h , k )
r
222 kyhxr
Standard form for the equation of a circle :
- center ( h , k )- radius ( r )
![Page 2: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/2.jpg)
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
![Page 3: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/3.jpg)
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…
![Page 4: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/4.jpg)
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…
![Page 5: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/5.jpg)
Distance and Circles
( h , k )
r
22 3436 yx
Find the center and radius of the circle whose equations is :
![Page 6: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/6.jpg)
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
![Page 7: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/7.jpg)
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
![Page 8: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/8.jpg)
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 )
![Page 9: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/9.jpg)
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 )
![Page 10: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/10.jpg)
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
![Page 11: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/11.jpg)
Distance and Circles
Completing the square – forcing an expression into a perfect square
trinomial
![Page 12: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/12.jpg)
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
![Page 13: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/13.jpg)
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.
![Page 14: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/14.jpg)
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.
![Page 15: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/15.jpg)
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 #
![Page 16: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/16.jpg)
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
![Page 17: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/17.jpg)
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
![Page 18: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/18.jpg)
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
![Page 19: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/19.jpg)
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…
![Page 20: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/20.jpg)
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
![Page 21: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/21.jpg)
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
![Page 22: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/22.jpg)
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
![Page 23: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/23.jpg)
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
![Page 24: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/24.jpg)
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
![Page 25: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/25.jpg)
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
![Page 26: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/26.jpg)
EXAMPLE #2 : Find the center and radius of a circle defined by the equation :
010302055 22 yxyx
![Page 27: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/27.jpg)
EXAMPLE #2 : Find the center and radius of a circle defined by the equation :
010302055 22 yxyx
![Page 28: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/28.jpg)
EXAMPLE #2 : Find the center and radius of a circle defined by the equation :
010302055 22 yxyx
![Page 29: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/29.jpg)
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
![Page 30: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/30.jpg)
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
![Page 31: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/31.jpg)
EXAMPLE #2 : Find the center and radius of a circle defined by the equation :
010302055 22 yxyx
026422 yxyx
![Page 32: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/32.jpg)
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…
![Page 33: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/33.jpg)
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…
![Page 34: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/34.jpg)
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…
![Page 35: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/35.jpg)
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
![Page 36: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/36.jpg)
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
![Page 37: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/37.jpg)
![Page 38: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/38.jpg)
![Page 39: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/39.jpg)
![Page 40: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/40.jpg)
EXAMPLE #3 : Find the center and radius of a circle defined by the equation :
047121822 yxyx
![Page 41: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/41.jpg)
EXAMPLE #3 : Find the center and radius of a circle defined by the equation :
047121822 yxyx
![Page 42: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/42.jpg)
EXAMPLE #3 : Find the center and radius of a circle defined by the equation :
047121822 yxyx
![Page 43: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/43.jpg)
EXAMPLE #3 : Find the center and radius of a circle defined by the equation :
047121822 yxyx
![Page 44: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/44.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :
![Page 45: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/45.jpg)
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 :
![Page 46: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/46.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :
![Page 47: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/47.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :
![Page 48: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/48.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :
![Page 49: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/49.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :
![Page 50: Distance and Circles ( h, k ) r Standard form for the equation of a circle : - center ( h, k ) - radius ( r )](https://reader035.vdocuments.us/reader035/viewer/2022062301/5697bf9a1a28abf838c91f68/html5/thumbnails/50.jpg)
EXAMPLE #2 : Find the equation of the circle whose center is ( 4 , 5 ) and goes
thru the coordinate ( - 1 , 6 ) :