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
- Which of the following best describes all points in a plane that are 5 inches from a given point in the plane?
- A circle with a 5 inch radius
- A circle with a 5 inch diameter
- A circle with a 25 inch radius
- A rectangle with 5 inch sides
- A sphere with a 5 inch diameter
- What is the center of the circle given by the equation (x – 8)2 + (y + 2)2 = 49 ?
- (8, 2)
- (7, 49)
- (8, –2)
- (7, –2)
- (8, –7)
- 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?
- (x – 5)2 – (y – 3)2 = 6
- (x + 5)2 + (y + 3)2 = 6
- (x – 5)2 + (y – 3)2 = 6
- (x – 5)2 + (y + 3)2 = 36
- (x + 5)2 – (y + 3)2 = 36
- 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?
- x2 + y2 = 1
- x2 – y2 = 1
- (x – 1)2 + (y – 4)2 = 2
- (x – 1)2 + (y – 4)2 = 1
- (x + 1)2 + (y + 4)2 = 1
- 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?
- x2 + y2 = 3
- (x – 3)2 – (y – 3)2 = 9
- (x – 3)2 + (y – 3)2 = 3
- (x – 3)2 + (y – 3)2 = 9
- (x + 3)2 + (y + 3)2 = 9
- A The question describes a circle. That's what a circle is, it's all the points in a plane that are a certain distance from a center point. Remember that radius is the distance from the center to the circle, and the diameter is twice that distance, i.e., twice the radius.
- H The center is (h, k) based on the equation (x – h)2 + (y – k)2 = r2. So match up the given equation to (x – h)2 + (y – k)2 = r2, and determine h and k. For (x – 8)2 + (y + 2)2 = 49, h = 8 and k = –2. Notice that we have (y + 2) but the formula is for (y – k), so we switch the sign of 2 to get –2. So, the center is (8, –2).
- D We just need to apply the info we are given into the equation for a circle, which is (x – h)2 + (y – k)2 = r2. Since the center is (5, –3) and the radius is 6, h = 5, k = –3, and r = 6. So D is the answer. Notice that we can easily get this question with process of elimination. For example, since r = 6, the number to the right of the equals sign, r2, must equal 36, not 6.
- J When a picture is described but not shown, draw it. The circle has center (1, 4). Since it is tangent to the y axis, our sketch shows that it has a radius of 1. So h = 1, k = 4, and r = 1. Choice J works.
- D When a picture is described but not shown, draw it. The sketch shows that the center must be (3, 3) and that the radius is therefore 3. So h = 3, k = 3, and r = 3. Choice D works. Watch out for choice B, which looks okay, but has a minus sign instead of a plus sign between the parentheses! That's why you've got to cut out the flash cards at the end of this book and memorize the equation for a circle.
Go to: Tip #40
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