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Sagot :
EQUATION OF A CIRCLE
[tex]\red{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]
Center-radius Form:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
General Form:
[tex]x^2+y^2+Cx+Dy+E=0[/tex]
☞ The given equations are in center-radius (standard) form. To change it to its general form, we will expand the equations and simplify.
[tex]\\[/tex]
1. (x + 4)² + (y – 7)² = 100
[tex] \tt (x + 4) ^{2} + (y - 7) ^{2} = 100 \\ \tt {x}^{2} + 8x + 16 + {y}^{2} - 14y + 49 = 100 \\ \tt {x}^{2} + 8x + {y}^{2} - 14y + 65 = 100 \\ \tt {x}^{2} + {y}^{2} + 8x - 14y + 65 = 100 \\ \tt {x}^{2} + {y}^{2} + 8x - 14y + 65 - 100 = 0 \\ \boxed{ \red{\tt {x}^{2} + {y}^{2} + 8x - 14y - 35 = 0}}[/tex]
[tex]\\[/tex]
2. (x - 1)² + (y - 4) ²= 64
[tex] \tt(x - 1) ^{2} + {(y - 4)}^{2} = 64 \\ \tt {x}^{2} - 2x + 1 + {y}^{2} - 8y + 16 = 64 \\ \tt {x}^{2} - 2x + {y}^{2} - 8y + 17 = 64 \\ \tt {x}^{2} + {y}^{2} - 2x - 8y + 17 = 64 \\ \tt {x}^{2} + {y}^{2} - 2x - 8y + 17- 64 = 0 \\ \boxed{ \red{ \tt {x }^{2} + {y}^{2} - 2x - 8y - 47 = 0 }}[/tex]
[tex]\\[/tex]
3. (x - 2)² + (y - 1)² = 11²
[tex] \tt{(x - 2)}^{2} + {(y - 1)}^{2} = {11}^{2} \\ \tt {(x - 2)}^{2} + {(y - 1)}^{2} = 121 \\ \tt {x}^{2} - 4x + 4 + {y}^{2} - 2y + 1 = 121 \\ \tt {x}^{2} - 4x + {y}^{2} - 2y + 5 = 121 \\ \tt {x}^{2} + {y}^{2} - 4x - 2y + 5 = 121 \\ \tt {x}^{2} + {y}^{2} - 4x - 2y + 5 - 121 = 0 \\ \boxed{ \red{ \tt {x}^{2} + {y}^{2} - 4x - 2y - 116 = 0 }}[/tex]
[tex]\\[/tex]
4. (x + 1)² + (y + 2)² = 25
[tex] \tt(x + 1 {)}^{2} + (y + 2 {)}^{2} = 25 \\ \tt {x}^{2} + 2x + 1 + {y}^{2} + 4y + 4 = 25 \\ \tt {x}^{2} + 2x + {y}^{2} + 4y + 5 = 25 \\ \tt {x}^{2} + {y}^{2} + 2x + 4y + 5 = 25 \\ \tt {x}^{2} + {y}^{2} + 2x + 4y + 5 - 25 = 0 \\ \boxed{ \red{ \tt {x}^{2} + {y}^{2} + 2x + 4y - 20 = 0}}[/tex]
[tex]\\[/tex]
5. (x - 2)² + (y - 4)² = 72
[tex] \tt(x - 2 {)}^{2} + (y - 4) ^{2} = 72 \\ \tt {x}^{2} - 4x + 4 + {y}^{2} - 8y + 16 = 72 \\ \tt {x}^{2} - 4x + {y}^{2} - 8y + 20 = 72 \\ \tt {x}^{2} + {y}^{2} - 4x - 8y + 20 = 72 \\ \tt {x}^{2} + {y}^{2} - 4x - 8y + 20 - 72 = 0 \\ \boxed{ \red{ \tt {x}^{2} + {y}^{2} - 4x - 8y - 52 = 0 }}[/tex]
[tex]\\[/tex]
[tex]\small{\textsf{Red text are the answers}}[/tex]
[tex]\red{\overline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}}[/tex]
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