Problem 1 Writing the Equation of a Circle

What is the standard equation of the circle with center ( 5 , − 2 ) open 5 comma negative 2 close  and radius 7?

( x − h ) 2 + ( y − k ) 2 = r 2 Use the standard form of an equation of a circle . ( x − 5 ) 2 + [ y − ( − 2 ) ] 2 = 7 2 Substitute  ( 5 , − 2 ) for  ( h , k )  and  7  for  r . ( x − 5 ) 2 + ( y + 2 ) 2 = 49 Simplify . table with 3 rows and 3 columns , row1 column 1 , open x minus h close squared . plus . open y minus k close squared , column 2 equals , r squared , column 3 cap usethestandardformofanequationofacircle . . , row2 column 1 , open x minus 5 close squared . plus . left bracket y minus open negative 2 close right bracket squared , column 2 equals , 7 squared , column 3 cap substitute . open 5 comma negative 2 close , for , open h comma k close , and , 7 , for , r . , row3 column 1 , open x minus 5 close squared . plus . open y plus 2 close squared , column 2 equals 49 , column 3 cap simplify , . , end table

✓ Got It?

1. What is the standard equation of each circle?
   1. center (3, 5); radius 6
   2. center ( − 2 , − 1 ) ; open minus , 2 comma negative 1 close semicolon  radius 2 square root of 2

Problem 2 Using the Center and a Point on a Circle

What is the standard equation of the circle with center ( 1 , − 3 ) open 1 comma negative 3 close  that passes through the point (2, 2)?

• Step 1 Use the Distance Formula to find the radius.

  r = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Use the Distance Formula .   = ( 1 − 2 ) 2 + ( − 3 − 2 ) 2 Substitute  ( 1 , − 3 )  for  ( x 2 , y 2 ) and  ( 2 , 2 )  for  ( x 1 , y 1 ) .   = ( − 1 ) 2 + ( − 5 ) 2 Simplify .   = 26   table with 4 rows and 3 columns , row1 column 1 , r , column 2 equals . square root of open . x sub 2 , minus , x sub 1 . close squared . plus . open . y sub 2 , minus , y sub 1 . close squared end root , column 3 cap usethecap distancecap formula . . , row2 column 1 , , column 2 equals . square root of open , 1 minus 2 , close squared . plus . open , negative 3 minus 2 , close squared end root , column 3 table with 2 rows and 1 column , row1 column 1 , cap substitute . open , 1 comma negative 3 , close . for . open . x sub 2 , comma , y sub 2 . close , row2 column 1 , and . open , 2 comma 2 , close . for . open . x sub 1 , comma , y sub 1 . close . . , end table , row3 column 1 , , column 2 equals . square root of open , negative 1 , close squared . plus . open , negative 5 , close squared end root , column 3 cap simplify , . , row4 column 1 , , column 2 equals square root of 26 , column 3 , end table

  

• Step 2 Use the radius and the center to write an equation.

  ( x − h ) 2 + ( y − k ) 2 = r 2 Use the standard form of an equation of a circle . ( x − 1 ) 2 + [ y − ( − 3 ) ] 2 = ( 26 ) 2 Substitute  ( 1 , − 3 )  for  ( h , k )  and  26  for  r . ( x − 1 ) 2 + ( y + 3 ) 2 = 26 Simplify . table with 3 rows and 3 columns , row1 column 1 , open , x minus h , close squared . plus . open , y minus k , close squared , column 2 equals , r squared , column 3 cap usethestandardformofanequationofacircle . . , row2 column 1 , open , x minus 1 , close squared . plus . left bracket . y minus . open , negative 3 , close . right bracket squared , column 2 equals . open square root of 26 close squared , column 3 cap substitute . open , 1 comma negative 3 , close . for . open , h comma k , close . and , square root of 26 , for , r . , row3 column 1 , open , x minus 1 , close squared . plus . open , y plus 3 , close squared , column 2 equals 26 , column 3 cap simplify , . , end table

✓ Got It?

2. What is the standard equation of the circle with center (4, 3) that passes through the point ( − 1 , 1 ) ? open minus , 1 comma 1 close question mark