zad.1
Odległość punktu P = ( x₀ , y₀ ) od prostej o równaniu ogólnym Ax + By + C = 0 możemy obliczyć ze wzoru:
[tex]d=\dfrac{\mid Ax_{0} +By_{0} +C\mid}{\sqrt{A^{2} +B^{2} } }[/tex]
[tex]P=(4,1)~~\Rightarrow ~~x_{0} =4~~\land~~y_{0} =1\\\\y=-2x-1~~\Rightarrow~~2x+y+1=0~~\Rightarrow~~A=2,~~B=1,~~C=1\\\\Podstawiam ~~do~~wzoru:\\\\d=\dfrac{\mid 2\cdot 4 + 1\cdot 1 +1 \mid}{\sqrt{2^{2} +1^{2} } } \\\\d=\dfrac{\mid 8+1+1\mid }{\sqrt{4+1} } \\\\d=\dfrac{10}{\sqrt{5} } \\\\usuwam~~niewymiernosc~~z~~mianownika:\\\\d=\dfrac{10}{\sqrt{5} } \cdot \dfrac{\sqrt{5} }{\sqrt{5} } \\\\d=\dfrac{10\sqrt{5} }{5} \\\\d=2\sqrt{5} \\\\Odp:~~A.~~2\sqrt{5}[/tex]
zad.2
Rówanie o srodku S = ( a,b ) i promieniu okręgu r ( r >0 )ma postać:
[tex](x-a)^{2} +(y-b)^{2} =r^{2}[/tex]
[tex]S=(0,-5)~~\Rightarrow ~~a=0~~\land ~~b=-5\\\\r=\sqrt{3} ~~\Rightarrow~~r^{2} =3\\\\Podstawiam ~~do~~wzoru:\\\\(x-0)^{2} +(y-(-5) )^{2} =3\\\\x^{2} +(y+5)^{2} =3\\\\Odp:~~C.~~x^{2} +(y+5)^{2} =3[/tex]
