[latex]Rownanie okregu o srodku S(a; b) i promieniu r:\\(x-a)^2+(y-b)^2=r^2\\\4.\a) x^2+y^2-4+m^2=0\x^2+y^2=4-m^2\\S(0; 0); r^2=4-m^2 > 0\\4-m^2 > 0\m^2 < 4\m < 2 wedge m > -2\\min(-2; 2)[/latex] [latex]b) x^2+y^2-2x+4y+m=0\\x^2-2xcdot1+1^2-1^2+y^2+2ycdot2+2^2-2^2+m=0\\(x-1)^2+(y+2)^2-1-4+m=0\\(x-1)^2+(y+2)^2=5-m\\S(1;-2); r^2=5-m > 0\\5-m > 0\\m < 5\\mim(-infty; 5)[/latex] [latex]5.\a) x^2-2x+y^2+4y=m^2-2m+3\\x^2-2xcdot1+1^2-1^2+y^2+2ycdot2+2^2-2^2=m^2-2m+3\\(x-1)^2+(y+2)^2-1-4=m^2-2m+3\\(x-1)^2+(y+2)^2=m^2-2m+3+5\\(x-1)^2+(y+2)^2=m^2-2m+8\\S(1;-2); r^2=m^2-2m+8 > 0[/latex] [latex]m^2-2m+8 > 0\\Delta=(-2)^2-4cdot1cdot8=4-32=-28 < 0\\minmathbb{R}[/latex] [latex]b) x^2+4x+y^2-my+m^2-frac{1}{4}m-12=0\\x^2+2xcdot2+2^2-2^2+y^2-2ycdotfrac{1}{2}m+(frac{1}{2}m)^2-(frac{1}{2}m)^2+m^2-frac{1}{4}m-12=0\\(x+2)^2+(y-frac{1}{2}m)^2-4-frac{1}{4}m^2+m^2-frac{1}{4}m-12=0\\(x+2)^2+(y-frac{1}{2}m)^2=-frac{3}{4}m^2+frac{1}{4}m+16\\S(-2; frac{1}{2}m); r^2=-frac{3}{4}m^2+frac{1}{4}m+16 > 0[/latex] [latex]-frac{3}{4}m^2+frac{1}{4}m+16 > 0 /cdot4\\-3m^2+m+64 > 0\\Delta=1^2-4cdot(-3)cdot64=1+768=769; sqrtDelta=sqrt{769}\\m_1=frac{-1-sqrt{769}}{2cdot(-3)}=frac{1+sqrt{769}}{6}; m_2=frac{1-sqrt{769}}{6}\\ramiona paraboli skierowane w dol\\min(frac{1-sqrt{769}}{6}; frac{1+sqrt{769}}{6})[/latex]
