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Homework: Circles

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides homework problems related to circles.

Exercise 1

Write down the equation for a circle (*aka “All the Points Equidistant from a Given Point”) with center (–3,–6) and radius 9. (Although the wording is different, this is exactly like the problems you did on the in-class assignment.)

Exercise 2

Now, let’s take it the other way. ( x – 4 ) 2 + ( y + 8 ) 2 = 49 ( x – 4 ) 2 + ( y + 8 ) 2 =49 is the equation for a circle.

  • a. What is the center of the circle?
  • b. What is the radius?
  • c. Draw the circle.
  • d. Find two points on the circle (by looking at your drawing) and plug them into the equation to make sure they work. (Show your work!)

2 x 2 + 2 y 2 + 8 x + 24 y + 60 = 0 2 x 2 +2 y 2 +8x+24y+60=0 is also the equation for a circle. But in order to graph it, we need to put it into our canonical form ( x – h ) 2 + ( y – k ) 2 = r 2 ( x – h ) 2 + ( y – k ) 2 = r 2 . In order to do that, we have to complete the square… twice! Here’s how it looks.

Table 1
2 x 2 + 2 y 2 + 8 x + 24 y + 60 = 0 2 x 2 +2 y 2 +8x+24y+60=0 The original problem.
x 2 + y 2 + 4 x + 12 y + 30 = 0 x 2 + y 2 +4x+12y+30=0 Divide by the coefficient of x 2 x 2 and y 2 y 2
( x 2 + 4 x ) + ( y 2 + 12 y ) = –30 ( x 2 +4x)+( y 2 +12y)=–30 Collect x x and y y terms together, and bring the number to the other side.
( x 2 + 4 x + 4 ) + ( y 2 + 12 y + 36 ) = –30 + 4 + 36 ( x 2 +4x+4)+( y 2 +12y+36)=–30+4+36 Complete the square in both parentheses.
( x + 2 ) 2 + ( y + 6 ) 2 = 10 ( x + 2 ) 2 + ( y + 6 ) 2 =10 Done! The center is (–2,–6) and the radius is 1010.

Got it? Now you try!

Exercise 3

3 x 2 + 3 y 2 + 18 x + 30 y – 6 = 0 3 x 2 +3 y 2 +18x+30y–6=0

  • a. Complete the square—as I did above—to put this into the form: ( x – h ) 2 + ( y – k ) 2 = r 2 ( x – h ) 2 + ( y – k ) 2 = r 2 .
  • b. What are the center and radius of the circle?
  • c. Draw the circle.
  • d. Find two points on the circle (by looking at your drawing) and plug them into original equation to make sure they work. (Show your work!)

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