Problem 4 Finding the Center and Radius

What are the center and radius of the circle with the given equation?

1. ( x − 16 ) 2 + ( y + 9 ) 2 = 144 open bold italic x minus 16 close squared . plus . open bold italic y plus 9 close squared . equals 144

   ( x − 16 ) 2 + ( y − ( − 9 ) ) 2 = 12 2 Rewrite the equation in standard form. h = 16 k = − 9 r = 12 Find  h , k , and  r . table with 2 rows and 2 columns , row1 column 1 , open x minus 16 close squared . plus . open y minus open negative 9 close close squared . equals , 12 squared , column 2 cap rewritetheequationinstandardform. , row2 column 1 , h equals 16 k equals negative 9 r equals 12 , column 2 cap find , h comma k comma , and , r . , end table

   The center of the circle is ( 16 , − 9 ) . open 16 comma negative 9 close .  The radius is 12.

2. x 2 + y 2 + 8 x − 10 y = 8 bold italic x squared , plus , bold italic y squared , plus 8 bold italic x minus 10 bold italic y equals 8

   ( x 2 + 8 x ) + ( y 2 − 10 y ) = 8 Set up to complete the squares. ( x 2 + 8 x + 16 ) + ( y 2 − 10 y + 25 ) = 8 + 16 + 25 Complete the squares and balance the equation. ( x + 4 ) 2 + ( y − 5 ) 2 = 49 Simplify. ( x − ( − 4 ) ) 2 + ( y − 5 ) 2 = 7 2 Rewrite the equation in standard form. h = − 4 k = 5 r = 7 Find  h , k , and  r . table with 5 rows and 2 columns , row1 column 1 , open , x squared , plus 8 x close plus open , y squared , minus 10 y close equals 8 , column 2 cap setuptocompletethesquares. , row2 column 1 , open , x squared , plus 8 x plus 16 close plus open , y squared , minus 10 y plus 25 close equals 8 plus 16 plus 25 , column 2 cap completethesquaresandbalancetheequation. , row3 column 1 , open x plus 4 close squared . plus . open y minus 5 close squared . equals 49 , column 2 cap simplify. , row4 column 1 , open x minus open negative 4 close close squared . plus . open y minus 5 close squared . equals , 7 squared , column 2 cap rewritetheequationinstandardform. , row5 column 1 , h equals negative 4 k equals 5 r equals 7 , column 2 cap find , h comma k comma , and , r . , end table

   The center of the circle is ( − 4 , 5 ) . open negative 4 comma 5 close .  The radius is 7.

✓ Got It?

4. What are the center and radius of the circle with the given equation?
   1. ( x + 8 ) 2 + ( y + 3 ) 2 = 121 open x plus 8 close squared . plus . open y plus 3 close squared . equals 121
   2. x 2 + y 2 − 6 x + 14 y = 8 x squared , plus , y squared , minus 6 x plus 14 y equals 8

Problem 5 Graphing a Circle Using Center and Radius

What is the graph of ( x + 1 ) 2 + ( y − 3 ) 2 = 25 ? open bold italic x plus 1 close squared . plus . open bold italic y minus 3 close squared . equals 25 question mark

( x − ( − 1 ) ) 2 + ( y − 3 ) 2 = 5 2 Rewrite the equation in standard form. h = − 1 k = 3 r = 5 Identify  h , k  and  r . center:  ( − 1 , 3 )  radius:  r = 5 Find the center and the radius of the circle. Locate the center  ( − 1 , 3 ) . Draw a circle of radius  5 . table with 3 rows and 2 columns , row1 column 1 , open x minus open negative 1 close close squared . plus . open y minus 3 close squared . equals , 5 squared , column 2 cap rewritetheequationinstandardform. , row2 column 1 , h equals negative 1 k equals 3 r equals 5 , column 2 cap identify . h comma k , and , r . , row3 column 1 , centercolon , open negative 1 comma 3 close . radiuscolon . r equals 5 , column 2 table with 3 rows and 1 column , row1 column 1 , cap findthecenterandtheradiusofthecircle. , row2 column 1 , cap locatethecenter . open negative 1 comma 3 close . , row3 column 1 , cap drawacircleofradius . 5 . , end table , end table

✓ Got It?

5. What is the graph of ( x − 4 ) 2 + ( y + 2 ) 2 = 49 ? open x minus 4 close squared . plus . open y plus 2 close squared . equals 49 question mark