SAT Math Problems: Using the Standard Circle Equation

The following practice questions give you the equation of a circle and ask you to find its radius and center.

Practice questions

The equation of a circle in the xy-plane is shown here:

x² + y² + 6x – 4y = –9

1. What is the radius of the circle? A. 1 B. 2 C. 3 D. 4
2. What are the (x, y) coordinates of the center? A. (–3, 2) B. (–2, 3) C. (3, –2) D. (2, –3)

Answers and explanations

1. The correct answer is Choice (B). First convert the equation to the standard circle equation:
   
   where r is the radius of the circle. From the original equation, start by moving the x's and y's together:
   
   The x² + 6x tells you that (x + 3)² is part of the equation. FOIL this out to x² + 6x + 9. However, the x² + 6x is by itself on the left, so add 9 to both sides of the equation:
   
   Also, y² – 4y tells you that (y – 2)² is part of the equation, which FOILs out to y² – 4y + 4. However, the y² – 4y is by itself on the left, so add 4 to both sides, like this:
   
   To convert the circle to its standard form, factor x² + 6x + 9 into (x + 3)² and y² – 4y + 4 into (y – 2)², like this:
   
   Now the circle is in its familiar form, and r² = 4, so r = 2.
2. The correct answer is Choice (A). First convert the equation to the standard circle equation:
   
   where h is the x-coordinate and k is the y-coordinate of the center of the circle. From the original equation, start by moving the x's and y's together:
   
   The x² + 6x tells you that (x + 3)² is part of the equation. FOIL this out to x² + 6x + 9. However, the x² + 6x is by itself on the left, so add 9 to both sides of the equation:
   
   Also, y² – 4y tells you that (y – 2)² is part of the equation, which FOILs out to y² – 4y + 4. However, the y² – 4y is by itself on the left, so add 4 to both sides, like this:
   
   To convert the circle to its standard form, factor x² + 6x + 9 into (x + 3)² and y² – 4y + 4 into (y – 2)², like this:
   
   Now the circle is in its familiar form, where h = –3 and k = 2.