1. **Identify the problem:** Find the center and radius of the circles given by equations in general form. --- Part A: Convert each equation to standard form and identify the center and radius. (1) $x^{2}+y^{2} -4x -6y -12 = 0$ Group terms: $(x^{2} -4x) + (y^{2} -6y) = 12$ Complete the square: $$x^{2}-4x = (x-2)^{2} -4$$ $$y^{2}-6y = (y-3)^{2} -9$$ Rewrite equation: $$(x-2)^{2} -4 + (y-3)^{2} -9 = 12$$ Simplify: $$(x-2)^{2} + (y-3)^{2} = 25$$ Center $(h,k) = (2,3)$, radius $r = 5$. (2) $x^{2}+y^{2} +8x -10y +8 = 0$ Group terms: $(x^{2} +8x) + (y^{2} -10y) = -8$ Complete the square: $$(x+4)^{2} -16 + (y-5)^{2} -25 = -8$$ Simplify: $$(x+4)^{2} + (y-5)^{2} = 33$$ Center $(-4,5)$, radius $ = oot{33}$ (approx 5.744). (3) $x^{2}+y^{2} +2x +6y -15 = 0$ Group terms: $$(x^{2} +2x) + (y^{2} +6y) = 15$$ Complete the square: $$(x+1)^{2} -1 + (y+3)^{2} -9 = 15$$ Simplify: $$(x+1)^{2} + (y+3)^{2} = 25$$ Center $(-1,-3)$, radius $5$. (4) $x^{2}+y^{2} -10x +4y +20 = 0$ Group terms: $$(x^{2} -10x) + (y^{2} +4y) = -20$$ Complete the square: $$(x-5)^{2} -25 + (y+2)^{2} -4 = -20$$ Simplify: $$(x-5)^{2} + (y+2)^{2} = 9$$ Center $(5,-2)$, radius $3$. (5) $x^{2}+y^{2} +6x +4y +9 = 0$ Group terms: $$(x^{2} +6x) + (y^{2} +4y) = -9$$ Complete the square: $$(x+3)^{2} -9 + (y+2)^{2} -4 = -9$$ Simplify: $$(x+3)^{2} + (y+2)^{2} = 4$$ Center $(-3,-2)$, radius $2$. (6) $x^{2}+y^{2} -8x -6y +9 = 0$ Group terms: $$(x^{2} -8x) + (y^{2} -6y) = -9$$ Complete the square: $$(x-4)^{2} -16 + (y-3)^{2} -9 = -9$$ Simplify: $$(x-4)^{2} + (y-3)^{2} = 16$$ Center $(4,3)$, radius $4$. (7) $x^{2}+y^{2} +12x -2y +14 = 0$ Group terms: $$(x^{2} +12x) + (y^{2} -2y) = -14$$ Complete the square: $$(x+6)^{2} -36 + (y-1)^{2} -1 = -14$$ Simplify: $$(x+6)^{2} + (y-1)^{2} = 23$$ Center $(-6,1)$, radius $ oot{23}$ (approx 4.796). (8) $x^{2}+y^{2} -2x +4y -4 = 0$ Group terms: $$(x^{2} -2x) + (y^{2} +4y) = 4$$ Complete the square: $$(x-1)^{2} -1 + (y+2)^{2} -4 = 4$$ Simplify: $$(x-1)^{2} + (y+2)^{2} = 9$$ Center $(1,-2)$, radius $3$. (9) $x^{2}+y^{2} -6x -2y -15 = 0$ Group terms: $$(x^{2} -6x) + (y^{2} -2y) = 15$$ Complete the square: $$(x-3)^{2} -9 + (y-1)^{2} -1 = 15$$ Simplify: $$(x-3)^{2} + (y-1)^{2} = 25$$ Center $(3,1)$, radius $5$. (10) $x^{2}+y^{2} +10x +8y +12 = 0$ Group terms: $$(x^{2} +10x) + (y^{2} +8y) = -12$$ Complete the square: $$(x+5)^{2} -25 + (y+4)^{2} -16 = -12$$ Simplify: $$(x+5)^{2} + (y+4)^{2} = 29$$ Center $(-5,-4)$, radius $ oot{29}$ (approx 5.385). --- Part B: Equations of circles in standard form given centers and radii. (1) Center $(3,-2)$, $r=5$ Equation: $$(x-3)^{2} + (y+2)^{2} = 25$$ (2) Center $(-4,3)$, $r=6$ Equation: $$(x+4)^{2} + (y-3)^{2} = 36$$ (3) Center $(1,2)$, $r=4$ Equation: $$(x-1)^{2} + (y-2)^{2} = 16$$ (4) Center $(-3,-5)$, $r=7$ Equation: $$(x+3)^{2} + (y+5)^{2} = 49$$ (5) Center $(0,0)$, $r=9$ Equation: $$x^{2} + y^{2} = 81$$ (6) Center $(2,4)$, $r=3$ Equation: $$(x-2)^{2} + (y-4)^{2} = 9$$ (7) Center $(-5,1)$, $r=8$ Equation: $$(x+5)^{2} + (y-1)^{2} = 64$$ (8) Center $(6,-2)$, $r=5$ Equation: $$(x-6)^{2} + (y+2)^{2} = 25$$ (9) Center $(1,-6)$, $r=10$ Equation: $$(x-1)^{2} + (y+6)^{2} = 100$$ (10) Center $(-7,5)$, $r=4$ Equation: $$(x+7)^{2} + (y-5)^{2} = 16$$ --- Part C: Graphing circle guide 1. Plot the center $(h,k)$. 2. Draw points $r$ units up, down, left and right from center to form boundary. 3. Sketch smooth circle through those points. 4. Label center and radius. All circles follow the form $(x-h)^{2} + (y-k)^{2} = r^{2}$. --- Final notes: Part A had 10 problems, Part B had 10 rewritten equations, and Part C is a general guide. Total problems answered: 20.