Answered

(with solution)
Activity 1: Write the following equations of a circle to their general forms.
1. (x + 4)² + (y – 7)² = 100
2. (x - 1)² + (y - 4) ²= 64
3. (x - 2)² + (y - 1)² = 11²
4. (x + 1)² + (y + 2)² = 25
5. (x - 2)² + (y - 4)² = 72 ​ ​ ​

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✏️CIRCLE EQUATIONS

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[tex]\underline{\mathbb{DIRECTION:}}[/tex]

- Write the following equations of a circle to their general forms.

  • #1. (x + 4)² + (y - 7)² = 100
  • #2. (x - 1)² + (y - 4)² = 64
  • #3. (x - 2)² + (y - 1)² = 11²
  • #4. (x + 1)² + (y + 2)² = 25
  • #5. (x - 2)² + (y - 4)² = 72

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[tex]\underline{\mathbb{ANSWERS:}}[/tex]

[tex]\qquad\rm»\:\:1.\:\green{{x}^{2} + {y}^{2} + 8x - 14y - 35 = 0}[/tex]

[tex]\qquad\rm»\:\:2.\:\green{{x}^{2} + {y}^{2} - 2x - 8y - 47 = 0}[/tex]

[tex]\qquad\rm»\:\:3.\:\green{{x}^{2}+y^2-4x-2y-116=0}[/tex]

[tex]\qquad\rm»\:\:4.\:\green{{x}^{2}+y^2+2x+4y-20=0}[/tex]

[tex]\qquad\rm»\:\:5.\:\green{{x}^{2} + {y}^{2} - 4x - 8y - 52 = 0}[/tex]

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[tex]\underline{\mathbb{SOLUTIONS:}}[/tex]

- The general form of the circle equation is written as as.

  • [tex] {x}^{2} + {y}^{2} + Ax + By + C = 0[/tex]

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#1. (x + 4)² + (y - 7)² = 100

- Expand the square of the binomials then rearrange the rest of the terms.

  • [tex] \small {x}^{2} + 8x + 16 + (y - 7)^{2} - 100 = 0[/tex]

  • [tex] \small {x}^{2} + 8x + 16 + {y}^{2} - 14y + 49 - 100 = 0[/tex]

  • [tex] \small {x}^{2} + {y}^{2} + 8x - 14y +16 + 49 - 100 = 0[/tex]

  • [tex] {x}^{2} + {y}^{2} + 8x - 14y - 35 = 0[/tex]

[tex]\therefore[/tex] x² + y² + 8x - 14y - 35 = 0 is the general form of the circle equation.

[tex]\rm[/tex]

#2. (x - 1)² + (y - 4)² = 64

- Expand the square of the binomials then rearrange the rest of the terms.

  • [tex] {x}^{2} - 2x + 1 + (y - 4)^{2} - 64 = 0[/tex]

  • [tex] \small {x}^{2} - 2x + 1 + {y}^{2} - 8y + 16 - 64 = 0[/tex]

  • [tex] \small {x}^{2} + {y}^{2} - 2x - 8y + 1 + 16 - 64 = 0[/tex]

  • [tex] {x}^{2} + {y}^{2} - 2x - 8y - 47 = 0[/tex]

[tex]\therefore[/tex] x² + y² - 2x - 8y - 47 = 0 is the general form of the circle equation.

[tex]\rm[/tex]

#3. (x - 2)² + (y - 1)² = 11²

- Expand the square of the binomials then rearrange the rest of the terms.

  • [tex] {x}^{2} - 4x + 4 + (y - 1)^{2} = 121[/tex]

  • [tex] {x}^{2} - 4x + 4 + y^2-2y+1 - 121[/tex]

  • [tex] {x}^{2}+y^2-4x-2y+4+1- 121=0[/tex]

  • [tex] {x}^{2}+y^2-4x-2y-116=0[/tex]

[tex]\therefore[/tex] x² + y² - 4x - 2y - 116 = 0 is the general form of the circle equation.

[tex]\rm[/tex]

#4. (x + 1)² + (y + 2)² = 25

- Expand the square of the binomials then rearrange the rest of the terms.

  • [tex] {x}^{2}+2x+1+(y+2)^2-25=0[/tex]

  • [tex] {x}^{2}+2x+1+y^2+4y+4-25=0[/tex]

  • [tex] {x}^{2}+y^2+2x+4y+1+4-25=0[/tex]

  • [tex] {x}^{2}+y^2+2x+4y-20=0[/tex]

[tex]\therefore[/tex] x² + y² + 2x + 4y - 20 = 0 is the general form of the circle equation.

[tex]\rm[/tex]

#5. (x - 2)² + (y - 4)² = 72

- Expand the square of the binomials then rearrange the rest of the terms.

  • [tex] {x}^{2} - 4x + 4 + (y - 4)^{2} - 72 = 0[/tex]

  • [tex] \small {x}^{2} - 4x + 4 + {y}^{2} - 8y + 16 - 72 = 0[/tex]

  • [tex] \small {x}^{2} + {y}^{2} - 4x - 8y +4 + 16 - 72 = 0[/tex]

  • [tex] \small {x}^{2} + {y}^{2} - 4x - 8y - 52 = 0[/tex]

[tex]\therefore[/tex] x² + y² - 4x - 8y - 52 = 0 is the general form of the circle equation.

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