Tip #39 to Get a Top ACT Math Score (page 2)

I love this Skill. Yes, I do. And I'll tell you why. Almost every ACT has one question asking you to determine the equation of a circle. Nobody has this thing memorized, so the question is always ranked as a "medium," or even a "hard." But if you know the formula, it's totally easy! These are the best.

The formula for the 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. Memorize this and you will gain a point on your ACT score, guaranteed. Don't you love this book! In fact, I want you to take a study break right now to call two friends and recommend this book. Or, log onto Amazon.com and write a rave review expressing your gratitude and appreciation!

Let's look at this question:

Solution: The center is (4, 2), so h = 4 and k = 2. Plus, the radius is 5, so r = 5. Easy. Plug that into the equation for a circle, (x – h)² + (y – k)² = r², which you have memorized. So the answer is E, or (x – 4)² + (y – 2)² = 25. Easy points!

Correct answer: E

Example Problems

Medium

1. Which of the following best describes all points in a plane that are 5 inches from a given point in the plane?
   1. A circle with a 5 inch radius
   2. A circle with a 5 inch diameter
   3. A circle with a 25 inch radius
   4. A rectangle with 5 inch sides
   5. A sphere with a 5 inch diameter
2. What is the center of the circle given by the equation (x – 8)² + (y + 2)² = 49 ?
   6. (8, 2)
   7. (7, 49)
   8. (8, –2)
   10. (7, –2)
   11. (8, –7)
3. A circle in the standard (x, y) coordinate plane has center (5,–3) and radius 6 coordinate units. Which of the following is an equation of the circle?
   1. (x – 5)² – (y – 3)² = 6
   2. (x + 5)² + (y + 3)² = 6
   3. (x – 5)² + (y – 3)² = 6
   4. (x – 5)² + (y + 3)² = 36
   5. (x + 5)² – (y + 3)² = 36

Hard

4. Which of the following is an equation of a circle with center at (1, 4) and tangent to the y axis in the standard (x, y) coordinate plane?
   6. x² + y² = 1
   7. x² – y² = 1
   8. (x – 1)² + (y – 4)² = 2
   10. (x – 1)² + (y – 4)² = 1
   11. (x + 1)² + (y + 4)² = 1
5. A circle in the standard (x, y) coordinate plane is tangent to the x axis at 3 and tangent to the y axis at 3. Which of the following is an equation of the circle?
   1. x² + y² = 3
   2. (x – 3)² – (y – 3)² = 9
   3. (x – 3)² + (y – 3)² = 3
   4. (x – 3)² + (y – 3)² = 9
   5. (x + 3)² + (y + 3)² = 9