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Sagot :
✏️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]
[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••}[/tex]
[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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