Practice
Equations of Circles Practice
Fifty original questions on standard form, completing squares, coordinate graphs, axis tangency, diameter endpoints, and circle applications.
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Question 1
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 16.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 2
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 9.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 3
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 36.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 4
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 25.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 5
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 4.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 6
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 49.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 7
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 64.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 8
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 1.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 9
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 81.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 10
Explanation
Standard form is (x-h)²+(y-k)²=r², so reverse the visible binomial signs and take the positive square root of 100.
- Reasoning
Read the center from opposite signs and radius from the square root.
Question 11
Explanation
Substitute h=-3, k=2, and r=5 into (x-h)²+(y-k)²=r².
- Reasoning
Keep the center signs inside subtraction and square the radius once.
Question 12
Explanation
Substitute h=4, k=-1, and r=3 into (x-h)²+(y-k)²=r².
- Reasoning
Keep the center signs inside subtraction and square the radius once.
Question 13
Explanation
Substitute h=0, k=6, and r=4 into (x-h)²+(y-k)²=r².
- Reasoning
Keep the center signs inside subtraction and square the radius once.
Question 14
Explanation
Substitute h=-5, k=-2, and r=7 into (x-h)²+(y-k)²=r².
- Reasoning
Keep the center signs inside subtraction and square the radius once.
Question 15
Explanation
Substitute h=2, k=0, and r=8 into (x-h)²+(y-k)²=r².
- Reasoning
Keep the center signs inside subtraction and square the radius once.
Question 16
Explanation
Group variables and complete both squares to obtain (x-3)²+(y+2)²=25.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 17
Explanation
Group variables and complete both squares to obtain (x+4)²+(y-1)²=9.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 18
Explanation
Group variables and complete both squares to obtain (x-2)²+(y-5)²=16.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 19
Explanation
Group variables and complete both squares to obtain (x+1)²+(y+3)²=36.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 20
Explanation
Group variables and complete both squares to obtain (x-5)²+(y-2)²=4.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 21
Explanation
Group variables and complete both squares to obtain (x+3)²+(y+4)²=49.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 22
Explanation
Group variables and complete both squares to obtain x²+(y-3)²=25.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 23
Explanation
Group variables and complete both squares to obtain (x-4)²+y²=36.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 24
Explanation
Group variables and complete both squares to obtain (x+2)²+(y-6)²=9.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 25
Explanation
Group variables and complete both squares to obtain (x-6)²+(y+1)²=16.
- Reasoning
Half each linear coefficient, reverse its sign for the center, and compute r² carefully.
Question 26
Explanation
A circle with center (-3,3) and radius 2 has equation (x+3)²+(y-3)²=4.
- Reasoning
Read the center point and a radius before selecting the equation.
Question 27
Explanation
A circle with center (2,2) and radius 5 has equation (x-2)²+(y-2)²=25.
- Reasoning
Read the center point and a radius before selecting the equation.
Question 28
Explanation
A circle with center (-4,-2) and radius 3 has equation (x+4)²+(y+2)²=9.
- Reasoning
Read the center point and a radius before selecting the equation.
Question 29
Explanation
A circle with center (1,-4) and radius 4 has equation (x-1)²+(y+4)²=16.
- Reasoning
Read the center point and a radius before selecting the equation.
Question 30
Explanation
A circle with center (5,0) and radius 2 has equation (x-5)²+y²=4.
- Reasoning
Read the center point and a radius before selecting the equation.
Question 31
Explanation
The radius is the perpendicular center-to-y-axis distance, 4.
- Reasoning
For y-axis tangency use |h|; for x-axis tangency use |k|.
Question 32
Explanation
The radius is the perpendicular center-to-x-axis distance, 5.
- Reasoning
For y-axis tangency use |h|; for x-axis tangency use |k|.
Question 33
Explanation
The radius is the perpendicular center-to-y-axis distance, 6.
- Reasoning
For y-axis tangency use |h|; for x-axis tangency use |k|.
Question 34
Explanation
The radius is the perpendicular center-to-x-axis distance, 7.
- Reasoning
For y-axis tangency use |h|; for x-axis tangency use |k|.
Question 35
Explanation
The radius is the perpendicular center-to-x-axis distance, 4.
- Reasoning
For y-axis tangency use |h|; for x-axis tangency use |k|.
Question 36
Explanation
The midpoint is (-1,1). The squared distance from that midpoint to either endpoint is 16.
- Reasoning
Use midpoint first, then one endpoint-to-center distance squared.
Question 37
Explanation
The midpoint is (2,1). The squared distance from that midpoint to either endpoint is 25.
- Reasoning
Use midpoint first, then one endpoint-to-center distance squared.
Question 38
Explanation
The midpoint is (-3,2). The squared distance from that midpoint to either endpoint is 25.
- Reasoning
Use midpoint first, then one endpoint-to-center distance squared.
Question 39
Explanation
The midpoint is (4,5). The squared distance from that midpoint to either endpoint is 25.
- Reasoning
Use midpoint first, then one endpoint-to-center distance squared.
Question 40
Explanation
The midpoint is (-2,-1). The squared distance from that midpoint to either endpoint is 61.
- Reasoning
Use midpoint first, then one endpoint-to-center distance squared.
Question 41
Explanation
Completing squares gives (x-5)²+(y+2)²=49, so area is πr²=49π.
- Reasoning
If area is requested, use the completed equation's right side directly.
Question 42
Explanation
The y-axis distance is |-4|=4, so r²=16.
- Reasoning
Use the horizontal coordinate for y-axis tangency.
Question 43
Explanation
The vertical distance to the x-axis is 7, so r=7 and area=49π.
- Reasoning
Use |k| for x-axis tangency.
Question 44
Explanation
Completing squares gives (x+3)²+(y-4)²=25-c. Set 25-c=16, so c=9.
- Reasoning
Express r² in terms of the constant, then solve.
Question 45
Explanation
The center (2,-1) becomes (-2,2); the radius stays 5.
- Reasoning
Transform the center, not the visible binomial signs.
Question 46
Explanation
The midpoint is (-1,1). From center to endpoint, the changes are 6 and -4, so r²=36+16=52.
- Reasoning
Midpoint first; squared half-distance gives the right side.
Question 47
Explanation
Distance from (h,2) to (0,0) is 5: h²+4=25, so h²=21.
- Reasoning
Use the point-on-circle distance equation.
Question 48
Explanation
Standard form is (x+1)²+(y-6)²=16; center y is 6 and radius 4, so lowest y=2.
- Reasoning
Find center and radius, then subtract radius from k.
Question 49
Explanation
Completing squares gives (x-2)²+(y+5)²=-11, impossible for real x and y.
- Reasoning
Validate that r² is positive after completing squares.
Question 50
Explanation
Area 81π gives r=9. Moving vertically 9 from y=4 gives y=13 and y=-5.
- Reasoning
Convert area to radius, then hold x at the center coordinate.
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Questions to review
No mistakes this time. Excellent work.
- Question 1Read standard formEasy
- Question 2Read standard formEasy
- Question 3Read standard formEasy
- Question 4Read standard formEasy
- Question 5Read standard formEasy
- Question 6Read standard formEasy
- Question 7Read standard formEasy
- Question 8Read standard formEasy
- Question 9Read standard formEasy
- Question 10Read standard formEasy
- Question 11Write standard formEasy
- Question 12Write standard formEasy
- Question 13Write standard formEasy
- Question 14Write standard formEasy
- Question 15Write standard formEasy
- Question 16Completing the squareMedium
- Question 17Completing the squareMedium
- Question 18Completing the squareMedium
- Question 19Completing the squareMedium
- Question 20Completing the squareMedium
- Question 21Completing the squareMedium
- Question 22Completing the squareMedium
- Question 23Completing the squareMedium
- Question 24Completing the squareMedium
- Question 25Completing the squareMedium
- Question 26Coordinate graph to equationMedium
- Question 27Coordinate graph to equationMedium
- Question 28Coordinate graph to equationMedium
- Question 29Coordinate graph to equationMedium
- Question 30Coordinate graph to equationMedium
- Question 31Axis tangencyMedium
- Question 32Axis tangencyMedium
- Question 33Axis tangencyMedium
- Question 34Axis tangencyMedium
- Question 35Axis tangencyMedium
- Question 36Diameter endpointsMedium
- Question 37Diameter endpointsMedium
- Question 38Diameter endpointsMedium
- Question 39Diameter endpointsMedium
- Question 40Diameter endpointsMedium
- Question 41Area from equationHard
- Question 42Axis tangent equationHard
- Question 43Axis tangent areaHard
- Question 44Parameter in circle equationHard
- Question 45Circle translationHard
- Question 46Diameter equationHard
- Question 47Unknown center coordinateHard
- Question 48Circle extremaHard
- Question 49Circle feasibilityHard
- Question 50Coordinate extremaHard