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Question 1

A)Calculate the center and radius of the circle by completing the square of the equation x2+y2+12x−18y+17=0.(1 point)
Responses

center=(−12,18), radius=20
center= left parenthesis negative 12 comma 18 right parenthesis , radius=20

center=(12,−18), radius=20
center= left parenthesis 12 comma negative 18 right parenthesis , radius=20

center=(6,−9), radius=10
center= left parenthesis 6 comma negative 9 right parenthesis , radius=10

center=(−6,9), radius=10
center= left parenthesis negative 6 comma 9 right parenthesis , radius=10
Question 2
A)Using the equation x2+y2−8x+2y=8, calculate the center and radius of the circle by completing the square.(1 point)
Responses

center=(−4,1), radius=5
center= left parenthesis negative 4 comma 1 right parenthesis , radius=5

center=(2,−8), radius=25
center= left parenthesis 2 comma negative 8 right parenthesis , radius=25

center=(8,−2), radius=25
center= left parenthesis 8 comma negative 2 right parenthesis , radius=25

center=(4,−1), radius=5
center= left parenthesis 4 comma negative 1 right parenthesis , radius=5
Question 3
A)Regina is trying to sketch a circle on graph paper with the equation x2+y2−2x+22y+1=0. Calculate where she would need to graph the center and radius of the circle by completing the square.(1 point)
Responses

center=(1,−11), radius=11
center= left parenthesis 1 comma negative 11 right parenthesis , radius=11

center=(−2,22), radius=41
center= left parenthesis negative 2 comma 22 right parenthesis , radius=41

center=(−22,2), radius=41
center= left parenthesis negative 22 comma 2 right parenthesis , radius=41

center=(−1,11), radius=11
center= left parenthesis negative 1 comma 11 right parenthesis , radius=11
Question 4
A)A circle is represented by the equation x2+y2−16x−6y=−24. Complete the square to discover the center and radius of the circle.(1 point)
Responses

center=(−16,−6), radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8

center=(3,8), radius=7
center= left parenthesis 3 comma 8 right parenthesis , radius=7

center=(8,3), radius=7
center= left parenthesis 8 comma 3 right parenthesis , radius=7

center=(16,6), radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8
Question 5
A)Dedric is trying to graph a circle on a computer software program that has a grid. The equation he is graphing is x2+y2−20x−14y+5=0. Where would he place the center of the circle and what is its radius?(1 point)
Responses

center=(20,14), radius=5
center= left parenthesis 20 comma 14 right parenthesis , radius=5

center=(−20,−14), radius=8
center= left parenthesis negative 20 comma negative 14 right parenthesis , radius=8

center=(−10,−7), radius=5

1 answer

To solve each of the equations for the center and radius of the circles by completing the square, we will go through them one by one.

Question 1:

Given equation: \[ x^2 + y^2 + 12x - 18y + 17 = 0 \]

Rearranging: \[ x^2 + 12x + y^2 - 18y + 17 = 0 \] \[ x^2 + 12x + y^2 - 18y = -17 \]

Completing the square:

  • For \(x\): \[ x^2 + 12x \rightarrow (x + 6)^2 - 36 \]

  • For \(y\): \[ y^2 - 18y \rightarrow (y - 9)^2 - 81 \]

Now substituting back: \[ (x + 6)^2 - 36 + (y - 9)^2 - 81 = -17 \] \[ (x + 6)^2 + (y - 9)^2 - 117 = -17 \] \[ (x + 6)^2 + (y - 9)^2 = 100 \]

Center and radius:

  • Center: \((-6, 9)\)
  • Radius: \(10\)

Correct Response:

  • center=(−6,9), radius=10

Question 2:

Given equation: \[ x^2 + y^2 - 8x + 2y = 8 \]

Rearranging: \[ x^2 - 8x + y^2 + 2y = 8 \]

Completing the square:

  • For \(x\): \[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \]

  • For \(y\): \[ y^2 + 2y \rightarrow (y + 1)^2 - 1 \]

Now substituting back: \[ (x - 4)^2 - 16 + (y + 1)^2 - 1 = 8 \] \[ (x - 4)^2 + (y + 1)^2 - 17 = 8 \] \[ (x - 4)^2 + (y + 1)^2 = 25 \]

Center and radius:

  • Center: \((4, -1)\)
  • Radius: \(5\)

Correct Response:

  • center=(4,−1), radius=5

Question 3:

Given equation: \[ x^2 + y^2 - 2x + 22y + 1 = 0 \]

Rearranging: \[ x^2 - 2x + y^2 + 22y + 1 = 0 \]

Completing the square:

  • For \(x\): \[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]

  • For \(y\): \[ y^2 + 22y \rightarrow (y + 11)^2 - 121 \]

Now substituting back: \[ (x - 1)^2 - 1 + (y + 11)^2 - 121 + 1 = 0 \] \[ (x - 1)^2 + (y + 11)^2 - 121 = 0 \] \[ (x - 1)^2 + (y + 11)^2 = 121 \]

Center and radius:

  • Center: \((1, -11)\)
  • Radius: \(11\)

Correct Response:

  • center=(1,−11), radius=11

Question 4:

Given equation: \[ x^2 + y^2 - 16x - 6y = -24 \]

Rearranging: \[ x^2 - 16x + y^2 - 6y = -24 \]

Completing the square:

  • For \(x\): \[ x^2 - 16x \rightarrow (x - 8)^2 - 64 \]

  • For \(y\): \[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \]

Now substituting back: \[ (x - 8)^2 - 64 + (y - 3)^2 - 9 = -24 \] \[ (x - 8)^2 + (y - 3)^2 - 73 = -24 \] \[ (x - 8)^2 + (y - 3)^2 = 49 \]

Center and radius:

  • Center: \((8, 3)\)
  • Radius: \(7\)

Correct Response:

  • center=(8,3), radius=7

Question 5:

Given equation: \[ x^2 + y^2 - 20x - 14y + 5 = 0 \]

Rearranging: \[ x^2 - 20x + y^2 - 14y + 5 = 0 \]

Completing the square:

  • For \(x\): \[ x^2 - 20x \rightarrow (x - 10)^2 - 100 \]

  • For \(y\): \[ y^2 - 14y \rightarrow (y - 7)^2 - 49 \]

Now substituting back: \[ (x - 10)^2 - 100 + (y - 7)^2 - 49 + 5 = 0 \] \[ (x - 10)^2 + (y - 7)^2 - 144 = 0 \] \[ (x - 10)^2 + (y - 7)^2 = 144 \]

Center and radius:

  • Center: \((10, 7)\)
  • Radius: \(12\)

Note: There might be a miscalculation here as none of the provided options correspond accurately to the findings.

None of the responses for question 5 are correct based on my calculations.

To summarize the correct results:

  1. Question 1: center=(-6,9), radius=10
  2. Question 2: center=(4,−1), radius=5
  3. Question 3: center=(1,−11), radius=11
  4. Question 4: center=(8,3), radius=7
  5. Question 5: No correct responses available based on calculations.