a) x^2 + x + y^2 - 6 = 0 ( x + 1/2)^2 - 1/4 + ( y - 0)^2 - 6 = 0 ( x + 1/2)^2 + ( y - 0)^2 = 6 1/4 S = ( - 1/2; 0) r^2 = 6 1/4 = 25/4 r = 5/2 = 2,5 ===================== b) x^2 + y^2 - 4 y = 0 ( x - 0)^2 + ( y - 2)^2 - 4 = 0 (x - 0)^2 + ( y - 2)^2 = 4 S = ( 0; 2) r^2 = 4 => r = 2 ==================
Równania okręgów są podane w postaci ogólnej: [latex]x^2 + y^2 - 2ax - 2by + c = 0[/latex]. Środkiem okręgu jest punkt [latex]S = (a, b)[/latex], a promień ma długość [latex]r = sqrt{a^2 + b^2 -c}[/latex] a) [latex]x^2 + x + y^2 - 6 = 0[/latex] [latex]x^2 + y^2 + x - 6 = 0[/latex] Stąd: [latex]- 2a = 1 /:(-2)[/latex] [latex]a = - frac{1}{2}[/latex] [latex]- 2b = 0 /:(-2)[/latex] [latex]b = 0[/latex] Zatem: [latex]S = (- frac{1}{2}; 0)[/latex] [latex]r = sqrt{(-frac{1}{2})^2 + 0^2 -(- 6)} = sqrt{frac{1}{4}+ 6} = sqrt{6frac{1}{4}} = sqrt{frac{25}{4}} = frac{5}{2} = 2frac{1}{2}[/latex] Odp. S = (- ½; 0), r = 2½ b) [latex]x^2 + y^2 - 4y = 0[/latex] Stąd: [latex]- 2a = 0 /:(-2)[/latex] [latex]a = 0[/latex] [latex]- 2b = -4 /:(-2)[/latex] [latex]b = 2[/latex] Zatem: [latex]S = (0; 2)[/latex] [latex]r = sqrt{0^2 + 2^2 - 0} = sqrt{4} = 2[/latex] Odp. S = (0; 2), r = 2
