Geometry Quiz: Derive The Equation Of A Circle
20 questions · exam conditions
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Derive The Equation Of A CircleQuestion 1 of 20

What is the equation of the circle with center (−2,−1)(-2,-1) and radius 33? (Justify using that all points on the circle are the same distance from the center.)

(x+2)2+(y+1)2=3(x+2)^2+(y+1)^2=3
(x−2)2+(y−1)2=9(x-2)^2+(y-1)^2=9
(x+2)2+(y+1)2=9(x+2)^2+(y+1)^2=9
x2+y2=9x^2+y^2=9
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Geometry Quiz

Geometry Quiz: Derive The Equation Of A Circle

Practice Derive The Equation Of A Circle in Geometry with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Derive The Equation Of A Circle, giving you a quick way to practice the rules, question types, and explanations that matter most for Geometry.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

What is the equation of the circle with center (−2,−1)(-2,-1) and radius 33? (Justify using that all points on the circle are the same distance from the center.)

  1. (x+2)2+(y+1)2=3(x+2)^2+(y+1)^2=3
  2. (x−2)2+(y−1)2=9(x-2)^2+(y-1)^2=9
  3. (x+2)2+(y+1)2=9(x+2)^2+(y+1)^2=9 (correct answer)
  4. x2+y2=9x^2+y^2=9
Explanation: To write a circle's equation given its center and radius, we apply the geometric definition directly. A circle is the set of all points that are the same distance from a fixed center point. If the center is (-2, -1) and radius is 3, then any point (x, y) on the circle satisfies the distance formula: √[(x-(-2))² + (y-(-1))²] = 3. Squaring both sides gives (x+2)² + (y+1)² = 9. The standard form (x-h)² + (y-k)² = r² makes the center and radius immediately visible. A common error involves the signs: since we have center (-2, -1), the equation uses (x-(-2)) = (x+2) and (y-(-1)) = (y+1). When given center and radius, think of the distance relationship first, then translate to algebra.

Question 2

A circle has the equation x2+y2−6x+8y−11=0x^2 + y^2 - 6x + 8y - 11 = 0. What is the equation of the circle that has the same radius but is centered at the origin?

  1. x2+y2=36x^2 + y^2 = 36 (correct answer)
  2. x2+y2=6x^2 + y^2 = 6
  3. x2+y2=64x^2 + y^2 = 64
  4. x2+y2=100x^2 + y^2 = 100
Explanation: Complete the square for the given equation: x2−6x+y2+8y=11x^2 - 6x + y^2 + 8y = 11. For x: x2−6x=(x−3)2−9x^2 - 6x = (x-3)^2 - 9. For y: y2+8y=(y+4)2−16y^2 + 8y = (y+4)^2 - 16. Substituting: (x−3)2−9+(y+4)2−16=11(x-3)^2 - 9 + (y+4)^2 - 16 = 11, which gives (x−3)2+(y+4)2=36(x-3)^2 + (y+4)^2 = 36. The radius is 36=6\sqrt{36} = 6. A circle centered at the origin with the same radius has equation x2+y2=36x^2 + y^2 = 36. Choice B uses the radius instead of radius squared. Choices C and D are incorrect calculations of the radius squared.

Question 3

Which statement correctly identifies the center and radius of the circle given by the equation x2+y2−6x+4y−12=0x^2+y^2-6x+4y-12=0? (Use completing the square to interpret the geometry.)

  1. Center (3,−2)(3,-2) and radius 55 (correct answer)
  2. Center (−3,2)(-3,2) and radius 55
  3. Center (3,−2)(3,-2) and radius 2525
  4. Center (0,0)(0,0) and radius 55
Explanation: This problem requires deriving the center and radius from the general form of a circle equation. A circle is defined as all points equidistant from its center. To interpret x² + y² - 6x + 4y - 12 = 0, we complete the square for both variables to reach standard form (x-h)² + (y-k)² = r². For x terms: x² - 6x = (x-3)² - 9; for y terms: y² + 4y = (y+2)² - 4. Substituting gives (x-3)² + (y+2)² - 9 - 4 - 12 = 0, which simplifies to (x-3)² + (y+2)² = 25. This reveals center (3, -2) and radius √25 = 5. Students often mistake r² for r, thinking the radius is 25 instead of 5. Remember: completing the square transforms the distance relationship into readable geometric parameters.

Question 4

Which expression represents all points (x,y)(x,y) that are a fixed distance 55 from the center (−3,4)(-3,4)? (Use the distance formula.)

  1. (x+3)2+(y−4)2=5(x+3)^2+(y-4)^2=5
  2. (x+3)2+(y−4)2=25(x+3)^2+(y-4)^2=25 (correct answer)
  3. (x−3)2+(y+4)2=25(x-3)^2+(y+4)^2=25
  4. x2+y2=25x^2+y^2=25
Explanation: This problem directly applies the definition of a circle to derive its equation. A circle is the set of all points at a constant distance from a center point. Using the distance formula, if (x, y) is distance 5 from center (-3, 4), then √[(x-(-3))² + (y-4)²] = 5. Squaring both sides gives (x+3)² + (y-4)² = 25. The standard form (x-h)² + (y-k)² = r² clearly shows center (h, k) = (-3, 4) and radius² = 25. Students often confuse r with r², writing 5 instead of 25 on the right side, which would incorrectly suggest radius = √5. When translating from geometric description to algebraic equation, always square the given radius to match the squared distance formula.

Question 5

Which statement correctly identifies the center and radius of the circle given by x2+y2+4x−10y+13=0x^2 + y^2 + 4x - 10y + 13 = 0? Rewrite the equation in standard form to interpret it geometrically.

  1. Center (2,−5)(2,-5) and radius 44
  2. Center (−2,5)(-2,5) and radius 44 (correct answer)
  3. Center (−2,5)(-2,5) and radius 1616
  4. Center (0,0)(0,0) and radius 44
Explanation: The skill here is deriving the equation of a circle. A circle is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a fixed point, called the center. The distance formula applies implicitly when completing the square on the given equation. This reveals the form (x+2)2+(y−5)2=16(x + 2)^2 + (y - 5)^2 = 16, connecting to center (−2,5)(-2,5) and radius 44. This justifies choice B as the geometric interpretation. A misconception is failing to square the half-coefficients correctly, leading to wrong radius like 1616 instead of 44 in choice C. Always think about the geometric distance before algebra to verify the completed form.

Question 6

Which equation follows from the geometric definition of a circle with center (−2,1)(-2,1) that passes through the point (1,5)(1,5)?

Use the idea that every point on the circle is the same distance from the center.

  1. (x+2)2+(y−1)2=25(x+2)^2+(y-1)^2=25 (correct answer)
  2. (x−2)2+(y+1)2=25(x-2)^2+(y+1)^2=25
  3. (x+2)2+(y−1)2=5(x+2)^2+(y-1)^2=5
  4. x2+y2=25x^2+y^2=25
Explanation: The skill here is deriving the equation of a circle from its center and a point it passes through. A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance formula applies by calculating the radius as the distance from the center (-2, 1) to (1, 5), which is (1+2)2+(5−1)2=9+16=5\sqrt{(1 + 2)^2 + (5 - 1)^2} = \sqrt{9 + 16} = 5. This connects algebraically to (x+2)2+(y−1)2=25(x + 2)^2 + (y - 1)^2 = 25, the standard form with center (-2, 1) and r2=25r^2 = 25. Option A is justified because it matches this derivation from the geometric definition. A distractor like option C uses r=5r = 5 without squaring to 25, misunderstanding the equation form. To transfer, always compute the distance to find the radius before forming the equation.

Question 7

Which equation follows from the geometric definition of a circle with center (−1,4)(-1,4) that passes through the point (2,0)(2,0)? (Use the distance formula to represent "all points a fixed distance from the center.")

  1. (x−1)2+(y+4)2=25(x-1)^2+(y+4)^2=25
  2. (x+1)2+(y−4)2=25(x+1)^2+(y-4)^2=25 (correct answer)
  3. (x+1)2+(y−4)2=5(x+1)^2+(y-4)^2=5
  4. x2+y2=25x^2+y^2=25
Explanation: The skill involves deriving the equation of a circle using its geometric properties. A circle is defined as the set of all points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. This definition directly translates to the distance formula, where the distance between any point (x,y) on the circle and the center (h,k) equals the radius r. Algebraically, this becomes (x - h)^2 + (y - k)^2 = r^2, connecting the form to the center and radius. Given the center (-1,4) and point (2,0), the radius is 5, so the equation is (x+1)^2 + (y-4)^2 = 25, as derived from the distance. A common distractor misconception is using the radius without squaring it, resulting in =5 instead of 25. To transfer this strategy, always think about the distance from the center before jumping into algebraic manipulations.

Question 8

Refer to the figure below. A circle is drawn on the coordinate plane with a horizontal radius shown from the center to a point on the circle. Which equation represents this circle?

  1. (x−3)2+(y−2)2=16(x-3)^2 + (y-2)^2 = 16 (correct answer)
  2. (x−3)2+(y−2)2=4(x-3)^2 + (y-2)^2 = 4
  3. (x+3)2+(y+2)2=16(x+3)^2 + (y+2)^2 = 16
  4. (x−7)2+(y−2)2=16(x-7)^2 + (y-2)^2 = 16
Explanation: The center is at (3,2)(3,2) and the radius extends to (7,2)(7,2), giving r=4r=4 and r2=16r^2=16. Choice B uses rr instead of r2r^2. Choice C reverses the signs of the center. Choice D confuses the endpoint of the radius with the center.

Question 9

Which statement correctly identifies the center and radius of the circle given by x2+y2+8x−2y−8=0x^2+y^2+8x-2y-8=0?

  1. Center (−4,1)(-4,1) and radius 55 (correct answer)
  2. Center (4,−1)(4,-1) and radius 55
  3. Center (−4,1)(-4,1) and radius 2525
  4. Center (0,0)(0,0) and radius 55
Explanation: This question requires completing the square to find a circle's center and radius from general form. A circle is defined as all points at a fixed distance from a center. Starting with x² + y² + 8x - 2y - 8 = 0, we rearrange and complete the square: (x² + 8x + 16) + (y² - 2y + 1) = 8 + 16 + 1. This gives (x+4)² + (y-1)² = 25, revealing center (-4,1) and radius √25 = 5. The transformation from general to standard form exposes the geometric meaning: the center coordinates are the opposites of what appears inside the parentheses. Students often forget that completing the square for (x + 8x) requires adding 16, not 64. Think of completing the square as revealing the hidden distance relationship.

Question 10

Which equation follows from the geometric definition of a circle for the set of all points (x,y)(x,y) that are a distance 66 from the center (1,−3)(1,-3)?

  1. (x−1)2+(y+3)2=6(x-1)^2+(y+3)^2=6
  2. (x−1)2+(y+3)2=36(x-1)^2+(y+3)^2=36 (correct answer)
  3. (x+1)2+(y−3)2=36(x+1)^2+(y-3)^2=36
  4. x2+y2=36x^2+y^2=36
Explanation: To derive the equation of a circle, we start with its fundamental definition. A circle consists of all points that maintain a constant distance from a fixed center point. Using the distance formula, if a point (x, y) is distance 6 from center (1, -3), then √[(x-1)² + (y-(-3))²] = 6. Squaring both sides eliminates the radical: (x-1)² + (y+3)² = 36. This standard form clearly shows the center at (1, -3) and radius² = 36, confirming radius = 6. A common mistake is writing r instead of r² on the right side, leading to equation (x-1)² + (y+3)² = 6. When deriving circle equations, always square the radius value to match the squared terms on the left.

Question 11

Which expression represents all points (x,y)(x,y) that are a fixed distance 66 from the center (−1,3)(-1,3)?

  1. (x+1)2+(y−3)2=6(x+1)^2+(y-3)^2=6
  2. (x+1)2+(y−3)2=36(x+1)^2+(y-3)^2=36 (correct answer)
  3. (x−1)2+(y+3)2=36(x-1)^2+(y+3)^2=36
  4. x2+y2=36x^2+y^2=36
Explanation: This question directly translates the geometric definition of a circle into its algebraic equation. A circle is the set of all points at a fixed distance from a center point. For points (x,y) that are distance 6 from center (-1,3), the distance formula gives √[(x-(-1))² + (y-3)²] = 6. Squaring both sides yields (x+1)² + (y-3)² = 36. Notice that the radius 6 becomes 36 when squared—this is crucial for the standard form. A common error is writing (x+1)² + (y-3)² = 6, forgetting that the equation uses r², not r. Remember: the equation of a circle always involves the square of the radius, reflecting the squared terms in the distance formula.

Question 12

The equation x2+y2−8x+6y+9=0x^2 + y^2 - 8x + 6y + 9 = 0 represents a circle. After completing the square, what are the coordinates of the center and the radius?

  1. Center: (4,−3)(4, -3), Radius: 44 (correct answer)
  2. Center: (4,−3)(4, -3), Radius: 22
  3. Center: (−4,3)(-4, 3), Radius: 44
  4. Center: (8,−6)(8, -6), Radius: 16\sqrt{16}
Explanation: Complete the square for both variables. For x terms: x2−8x=(x−4)2−16x^2 - 8x = (x-4)^2 - 16. For y terms: y2+6y=(y+3)2−9y^2 + 6y = (y+3)^2 - 9. Substituting: (x−4)2−16+(y+3)2−9+9=0(x-4)^2 - 16 + (y+3)^2 - 9 + 9 = 0, which simplifies to (x−4)2+(y+3)2=16(x-4)^2 + (y+3)^2 = 16. The center is (4,−3)(4, -3) and radius is 16=4\sqrt{16} = 4. Choice B has the correct center but wrong radius. Choice C has sign errors in the center coordinates. Choice D uses the original coefficients incorrectly.

Question 13

The endpoints of a diameter of a circle are A(−1,4)A(-1, 4) and B(7,−2)B(7, -2). Which equation represents this circle?

  1. (x−3)2+(y−1)2=25(x - 3)^2 + (y - 1)^2 = 25 (correct answer)
  2. (x−4)2+(y−1)2=25(x - 4)^2 + (y - 1)^2 = 25
  3. (x−3)2+(y−1)2=100(x - 3)^2 + (y - 1)^2 = 100
  4. (x−3)2+(y+1)2=25(x - 3)^2 + (y + 1)^2 = 25
Explanation: The center of the circle is the midpoint of the diameter: (−1+72,4+(−2)2)=(3,1)\left(\frac{-1+7}{2}, \frac{4+(-2)}{2}\right) = (3, 1). The radius is half the distance between A and B: r=12(7−(−1))2+(−2−4)2=1264+36=12100=5r = \frac{1}{2}\sqrt{(7-(-1))^2 + (-2-4)^2} = \frac{1}{2}\sqrt{64+36} = \frac{1}{2}\sqrt{100} = 5. Therefore r2=25r^2 = 25. The equation is (x−3)2+(y−1)2=25(x-3)^2 + (y-1)^2 = 25. Choice B has an incorrect x-coordinate for the center. Choice C uses the diameter squared instead of radius squared. Choice D has an incorrect y-coordinate for the center.

Question 14

Which statement correctly identifies the center and radius of the circle given by x2+y2+2x−12y+29=0x^2+y^2+2x-12y+29=0?

  1. Center (−1,6)(-1,6), radius 8\sqrt{8} (correct answer)
  2. Center (1,−6)(1,-6), radius 8\sqrt{8}
  3. Center (−1,6)(-1,6), radius 88
  4. Center (0,0)(0,0), radius 29\sqrt{29}
Explanation: The skill involves deriving the equation of a circle using its geometric properties. A circle is defined as the set of all points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. This definition directly translates to the distance formula, where the distance between any point (x,y) on the circle and the center (h,k) equals the radius r. Algebraically, this becomes (x - h)^2 + (y - k)^2 = r^2, with completing the square revealing these elements. For the given equation, completing the square yields center (-1,6) and radius √8, matching the geometric interpretation. A common distractor misconception is failing to square root for the radius, leading to 8 instead of √8. To transfer this strategy, always think about the distance from the center before jumping into algebraic manipulations.

Question 15

A circle is the set of all points a fixed distance from its center. What is the equation of the circle with center (2,−5)(2, -5) and radius 33?​

  1. (x−2)2+(y+5)2=9(x-2)^2+(y+5)^2=9 (correct answer)
  2. (x+2)2+(y−5)2=9(x+2)^2+(y-5)^2=9
  3. (x−2)2+(y+5)2=3(x-2)^2+(y+5)^2=3
  4. x2+y2=9x^2+y^2=9
Explanation: This question directly applies the definition of a circle to write its equation. A circle is the set of all points that are equidistant from a fixed center point, creating a perfect round shape. Using the distance formula, every point (x, y) on a circle with center (2, -5) and radius 3 satisfies √[(x - 2)² + (y - (-5))²] = 3. Squaring both sides to eliminate the square root gives us (x - 2)² + (y + 5)² = 9, which is the standard form of the circle equation. Notice how the center coordinates appear in the parentheses with opposite signs: center (2, -5) gives us (x - 2) and (y + 5). A common mistake is writing (y - 5) instead of (y + 5) when the y-coordinate is negative—always be careful with signs. To master circle equations, remember that they encode the simple geometric idea that every point maintains the same distance from the center.

Question 16

Which statement correctly identifies the center and radius of the circle given by x2+y2−6x+4y=12x^2+y^2-6x+4y=12? (Interpret by completing the square to reveal the geometric meaning.)

  1. Center (3,−2)(3,-2), radius 55 (correct answer)
  2. Center (−3,2)(-3,2), radius 55
  3. Center (3,−2)(3,-2), radius 2525
  4. Center (0,0)(0,0), radius 12\sqrt{12}
Explanation: The skill involves deriving the equation of a circle using its geometric properties. A circle is defined as the set of all points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. This definition directly translates to the distance formula, where the distance between any point (x,y) on the circle and the center (h,k) equals the radius r. Algebraically, this becomes (x - h)^2 + (y - k)^2 = r^2, and completing the square reveals the center and radius from the general form. For the given equation, completing the square yields center (3,-2) and radius 5, matching the geometric interpretation. A common distractor misconception is mistaking the radius for its square, leading to radius 25 instead of 5. To transfer this strategy, always think about the distance from the center before jumping into algebraic manipulations.

Question 17

Which expression represents all points (x,y)(x,y) that are a fixed distance 4 from the center (1,−3)(1,-3)?

Choose the equation that directly encodes "distance from (x,y)(x,y) to the center equals 4."

  1. (x+1)2+(y−3)2=16(x+1)^2+(y-3)^2=16
  2. (x−1)2+(y+3)2=16(x-1)^2+(y+3)^2=16 (correct answer)
  3. (x−1)2+(y+3)2=4(x-1)^2+(y+3)^2=4
  4. x2+y2=16x^2+y^2=16
Explanation: The skill here is deriving the equation of a circle based on its center and radius. A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance formula is applied by equating the distance from (x,y)(x, y) to the center (1,−3)(1, -3) to 4, giving (x−1)2+(y+3)2=4\sqrt{(x - 1)^2 + (y + 3)^2} = 4. Algebraically, this connects to the form (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, where h = 1, k = -3, and r2=16r^2 = 16. Option B is correct as it directly encodes this distance condition with (x−1)2+(y+3)2=16(x - 1)^2 + (y + 3)^2 = 16. A misconception in option C is using r = 4 without squaring it to get 16. For transfer, focus on the distance requirement before writing the algebraic equation.

Question 18

What is the equation of the circle with center (0,−2)(0,-2) and radius 33?

  1. (x−0)2+(y+2)2=3(x-0)^2+(y+2)^2=3
  2. (x−0)2+(y+2)2=9(x-0)^2+(y+2)^2=9 (correct answer)
  3. (x−0)2+(y−2)2=9(x-0)^2+(y-2)^2=9
  4. x2+y2=9x^2+y^2=9
Explanation: This question asks you to construct a circle equation from its defining properties. A circle is geometrically defined as all points at a fixed distance from a center point. Given center (0, -2) and radius 3, we apply the distance formula: for any point (x, y) on the circle, √[(x - 0)² + (y - (-2))²] = 3. Squaring both sides eliminates the square root, yielding (x - 0)² + (y + 2)² = 9. The standard form (x - h)² + (y - k)² = r² directly shows the center at (h, k) = (0, -2) and radius r = 3. Notice how y - (-2) simplifies to y + 2, reflecting the center's negative y-coordinate. A common mistake is writing 3 instead of 9 on the right side, forgetting that we need r² = 3² = 9 in the standard equation. When constructing circle equations, always verify that the radius squared appears on the right side.

Question 19

A circle is the set of all points a fixed distance from its center. What is the equation of the circle with center (−2,3)(-2, 3) and radius 44?​

  1. (x+2)2+(y−3)2=16(x+2)^2+(y-3)^2=16 (correct answer)
  2. (x−2)2+(y+3)2=16(x-2)^2+(y+3)^2=16
  3. (x+2)2+(y−3)2=4(x+2)^2+(y-3)^2=4
  4. x2+y2=16x^2+y^2=16
Explanation: This question asks you to derive the equation of a circle from its geometric properties. A circle is defined as the set of all points that are equidistant from a fixed center point. Using the distance formula, any point (x, y) on the circle must satisfy √[(x - h)² + (y - k)²] = r, where (h, k) is the center and r is the radius. Squaring both sides gives us the standard form: (x - h)² + (y - k)² = r². For a circle with center (-2, 3) and radius 4, we substitute to get (x - (-2))² + (y - 3)² = 4², which simplifies to (x + 2)² + (y - 3)² = 16. A common mistake is confusing the signs when the center has negative coordinates—remember that (x - h) becomes (x + 2) when h = -2. To avoid errors, always think of the distance relationship first before manipulating the algebra.

Question 20

Which statement correctly identifies the center and radius of the circle given by x2+y2+8x−10y=7?x^2+y^2+8x-10y=7?

Interpret the equation by rewriting it to show a fixed distance from a center.

  1. Center (4,−5)(4,-5) and radius 48\sqrt{48}
  2. Center (−4,5)(-4,5) and radius 48\sqrt{48} (correct answer)
  3. Center (−4,5)(-4,5) and radius 4848
  4. Center (0,0)(0,0) and radius 7\sqrt{7}
Explanation: The skill here is deriving the equation of a circle and extracting center and radius by rewriting it. A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance formula underlies the process, as completing the square reveals the form where distance to (h,k)(h, k) equals rr. Algebraically, rewriting x2+y2+8x−10y=7x^2 + y^2 + 8x - 10y = 7 gives (x+4)2+(y−5)2=48(x + 4)^2 + (y - 5)^2 = 48, so center (−4,5)(-4, 5) and r=48r = \sqrt{48}. Option B is correct because it matches this interpretation of fixed distance from the center. A misconception in option C is taking 48 as the radius without the square root. For transfer, interpret the equation through distance before completing the square.