Geometry Practice Test: Equations of Circles
- Equations of Circles
- 10 questions
- 13 minutes
These 10 questions use the standard form (x − h)² + (y − k)² = r² of a circle. Tip: the signs inside the parentheses are the opposite of the center coordinates.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
Which is the equation of the circle with center (3, −2) and radius 5?
Correct answer: C
Use (x − h)2 + (y − k)2 = r2 with h = 3, k = −2 and r2 = 25. The right side is the radius squared, not 5.
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Question 2
What are the center and radius of (x + 4)2 + (y − 1)2 = 49?
Correct answer: A
(x + 4) = (x − (−4)), so h = −4 and k = 1. The radius is 49 = 7.
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Question 3
What are the center and radius of x2 + y2 − 6x + 8y = 0?
Correct answer: D
Complete the square: (x2 − 6x + 9) + (y2 + 8y + 16) = 25, so (x − 3)2 + (y + 4)2 = 25. The center is (3, −4) and the radius is 5.
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Question 4
A circle has center (1, 2) and passes through the point (4, 6). What is its equation?
Correct answer: B
The radius is the distance from (1, 2) to (4, 6): 32 + 42 = 5, so r2 = 25. The last option wrongly uses the point on the circle as the center.
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Question 5
The endpoints of a diameter of a circle are (−2, 3) and (6, −3). What is the equation of the circle?
Correct answer: A
The center is the midpoint (2, 0). The diameter is 82 + 62 = 10, so the radius is 5 and r2 = 25. Using 10 as the radius gives 100.
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Question 6
Where is the point (5, 1) in relation to the circle (x − 2)2 + (y + 3)2 = 25?
Correct answer: D
Substitute: (5 − 2)2 + (1 + 3)2 = 9 + 16 = 25, which equals r2. So the point is exactly 5 units from the center and lies on the circle.
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Question 7
Which point lies on the circle x2 + y2 = 100?
Correct answer: C
62 + (−8)2 = 36 + 64 = 100. For (10, 10) the sum is 200, so that point is outside.
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Question 8
What is the area of the circle (x + 1)2 + (y − 5)2 = 18?
Correct answer: B
The equation already gives r2 = 18, and the area is πr2 = 18π. Squaring 18 again gives the trap 324π.
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Question 9
The circle x2 + y2 = 9 is translated 2 units right and 5 units down. What is the equation of the image?
Correct answer: D
The center moves from (0, 0) to (2, −5) and the radius stays 3, so the equation is (x − 2)2 + (y + 5)2 = 9.
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Question 10
Which line is tangent to the circle x2 + y2 = 25 at the point (3, 4)?
Correct answer: A
A tangent is perpendicular to the radius. The radius to (3, 4) has slope 43, so the tangent has slope −34 and passes through (3, 4): 3x + 4y = 25. The line 4x − 3y = 0 is the radius line itself.
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