Question

(UFRGS) a distância do ponto (2;m) a reta x-y=0 é √8. O valor de M é:​

Answer 1

Distância do ponto a reta

$\boxed{\boxed{\mathtt{d_{p,r}=\dfrac{|ax_{p}+by_{p}+c|}{\sqrt{{a}^{2}+{b}^{2}}}}}}$

$\mathtt{x_{p}~e~y_{p}}$ são coordenadas do ponto dado e a,b são os coeficientes da equação geral da reta.

P(2,m)

$\mathtt{r:1x-1y+0=0\to\,a=1~b=-1~c=0}$

$\mathtt{d_{p,r}=\dfrac{|ax_{p}+by_{p}+c|}{\sqrt{{a}^{2}+{b}^{2}}}}$

Substituindo temos:

$\mathtt{\sqrt{8}=\dfrac{|1.2-1.m+0|}{\sqrt{{(1)}^{2}+{(-1)}^{2}}}}$

$\mathtt{\sqrt{8}=\dfrac{|2-m|}{\sqrt{2}}}$

$\mathtt{|2-m|=\sqrt{2}.\sqrt{8}}\\\mathtt{|2-m|=\sqrt{16}}\\\mathtt{|2-m|=4}$

Definição de módulo

$\displaystyle\large\boxed{\boxed{|x|=\begin{cases}\mathtt{x~se~x\ge0}\\\mathtt{-x~se~x\textless0}\end{cases}}}$

$\mathtt{|2-m|=4}\\\mathtt{2-m=4~ou~2-m=-4}</p><p>\\\mathtt{m=2-4~ou~m=2+4}\\\mathtt{m=-2~ou~m=6}$

$\boxed{\boxed{\mathtt{Alternativa~~d}}}$