Temos que lembrar como fazer distância de ponto a ponto, que calculamos da seguinte forma:
[tex]\boxed{d = \sqrt{(X_{b}-X_{a})^{2} + (Y_{b}-Y_{a})^{2}}}[/tex]
Agora vamos substituir os valores:
[tex]d = \sqrt{(X_{b}-X_{a})^{2} + (Y_{b}-Y_{a})^{2}}
\\\\
\sqrt{(2-(m-1))^{2} + (-m-3)^{2}} = 6
\\\\
\sqrt{(2-m+1)^{2} + (-m-3)^{2}} = 6
\\\\
\sqrt{(3-m+)^{2} + (-m-3)^{2}} = 6
\\\\
elevamos \ ao \ quadrado \ para \ sumir \ com \ a \ raiz
\\\\
(\sqrt{(3-m)^{2} + (-m-3)^{2}})^{2} = 6^{2}
\\\\
(3-m)^{2} + (-m-3)^{2}} = 36
\\\\
9 - 6m + m^{2} + m^{2} + 6m + 9 = 36[/tex]
[tex]9 - 6m + m^{2} + m^{2} + 6m + 9 -36 = 0
\\\\
2m^{2} - 18 = 0
\\\\
\Delta = b^{2} - 4 \cdot a \cdot c
\\\\
\Delta = 0^{2} - 4 \cdot (2) \cdot (-18)
\\\\
\Delta = 0+144
\\\\
\Delta = 144[/tex]
[tex]2m^{2} - 18 = 0
\\\\
m = \frac{-b \pm \sqrt{\Delta}}{2 \cdot a}
\\\\
m = \frac{0 \pm \sqrt{144}}{2 \cdot 2}
\\\\
m = \frac{0 \pm 12}{4}
\\\\\\
m' = \frac{0 + 12}{4} = \frac{12}{4} = \boxed{3}
\\\\
m'' = \frac{0 - 12}{4} = \frac{-12}{4} = \boxed{-3}[/tex]
Vamos voltar e testar os valores. Em ambos, a distância obrigatoriamente deve dar 6.
[tex]\rightarrow m = 3
\\
a(m-1; 3) \ e \ b(2; -m)
\\
a(3-1; 3) \ e \ b(2; -3)
\\
a(2; 3) \ e \ b(2; -3)
\\\\
d = \sqrt{(X_{b}-X_{a})^{2} + (Y_{b}-Y_{a})^{2}}
\\\\
d = \sqrt{(2-2)^{2} + (-3-3)^{2}}
\\\\
d = \sqrt{0 + (-6)^{2}}
\\\\
d = \sqrt{36}
\\\\
\boxed{d = 6}[/tex]
Testando o segundo:
[tex]\rightarrow m = -3
\\
a(m-1; 3) \ e \ b(2; -m)
\\
a(-3-1; 3) \ e \ b(2; -(-3))
\\\
a(-4; 3) \ e \ b(2; 3)
\\\\
d = \sqrt{(X_{b}-X_{a})^{2} + (Y_{b}-Y_{a})^{2}}
\\\\
d = \sqrt{(2-(-4))^{2} + (3-3)^{2}}
\\\\
d = \sqrt{(2+4)^{2} + 0}
\\\\
d = \sqrt{(6)^{2}}
\\\\
d = \sqrt{36}
\\\\
\boxed{d = 6}[/tex]
[tex]\therefore \boxed{\boxed{m = 3} \ e \ \boxed{m = -3}}[/tex]
