b) [tex]x^{2}[/tex]-6x+[tex]y^{2}[/tex]-10y-2=0
([tex]x^{2}[/tex]-6x+9)+([tex]y^{2}[/tex]-10y+25)-9-5-2=0
[tex](x-3)^{2} + (y-5)^{2} = 16[/tex]
S1(3,5) r1=4
[tex]x^{2} -14x+y^{2}+8y+56=0[/tex]
[tex](x^{2} -14x+49)+(y^{2}+8y+16)-49-16+56=0\\[/tex]
[tex](x-7)^{2}+(y+4)^{2}=9[/tex]
S2(7,-4) r2=3
|S1S2| = [tex]\sqrt{(7-3)^{2} + (-4-5)^{2}}=\sqrt{16+81}=\sqrt{97}[/tex] r1+r2=7
|S1S2| > r1+r2
Okręgi rozłączne zewnętrznie
c) [tex]x^{2} -2x+y^{2}-8y-3=0[/tex]
[tex](x^{2} -2x+1)+(y^{2}-8y+16)-1-16-3=0[/tex]
[tex](x-1)^{2}+(y-4)^{2} = 20[/tex]
S1(1,4) r1=[tex]2\sqrt{5}[/tex]
[tex]x^{2} -14x+y^{2}+16y+33=0[/tex]
[tex](x^{2}-14x+49)+(y^{2}+16y+64)-49-64+33=0[/tex]
[tex](x-7)^{2}+(y+8)^{2}=80[/tex]
S2(7,-8) r2=[tex]4\sqrt{5}[/tex]
|S1S2| = [tex]\sqrt{36+144}=\sqrt{180}=6\sqrt{5}[/tex]
|S1S2| = r1+r2
Okręgi styczne zewnętrznie
d) [tex]x^{2} -16x+y^{2}+4y+31=0[/tex]
[tex](x^{2}-16x+64)+(y^{2} +4y+4)-64-4+31=0[/tex]
[tex](x-8)^{2}+(y+2)^{2} = 37[/tex]
S1(8,-2) r1=[tex]\sqrt{37}[/tex]
[tex]x^{2}-4x+y^{2}-8y-5=0[/tex]
[tex](x^{2}-4x+4)+(y^{2}-8y+16)-4-16-5=0[/tex]
[tex](x-2)^{2}+(y-4)^{2}=25[/tex]
S2(2,4) r2=5
|S1S2| = [tex]\sqrt{36+4} =\sqrt{40}[/tex]
Okręgi przecinające się
