Test Prep Thursday: Equations of Circles

Which of the following gives the equation for the circle in the standard (x,y) coordinate plane with a center at (4, 8) and a radius of 10 units?

1. (x-4)² + (y-8)² = 100
2. (x+4)² + (y+8)² = 100
3. (x-4)² + (y-8)² = 10
4. (x-4)² - (y-8)² = 100
5. (x+4)² + (y+8)² = 10

In order to solve this problem, we’ll need to know what the standard equation of a circle is! The standard equation of a circle is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle, and r is the radius. So, in our example, the correct answer would be (1). Make sense? Let’s try something a bit more challenging.

Which of the following gives the equation for the circle in the standard (x,y) coordinate plane with a center at (-3, -5) and a circumference of 12π?

1. (x-3)² + (y-5)² = 12
2. (x+3)² + (y+5)² = 12
3. (x-3)² + (y-5)² = 144
4. (x+3)² + (y+5)² = 36
5. (x-3)² + (y-5)² = 36

So let’s break this problem down. First, we’ll focus on the center of the circle. Remember: the standard equation of a circle is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle, and r is the radius. So, with a center of negative 3, negative 5, our equation becomes (x+3)² + (y+5)² = r², because x-(-3) = x+3, and y-(-5) = y+5. So, we can narrow down the possible answers to (2) and (4), as they are the only equations that correctly identify the center. Now, we need to figure out our radius. In the problem, we are told that the circumference of the circle is 12π units. Given that circumference is equivalent to 2πr, our radius must be 6! (12π/2π = 6). So, our answer is D) (x+3)² + (y+5)² = 36, because 6² = 36.

For more ACT and SAT practice problems, check back in every Thursday to stay sharp and prepare to master those tests!