Question

Determine os numeros de k, de modo que a distancia entre o ponto P(K,1) e a reta (r) x -y=2 seja √2.

Answer 1

Boa tarde!

Solução!

Vamos usar essa formula para determinarmos o valor de K.

$d(P,r)= \dfrac{|xP+yP+c|}{ \sqrt{a^{2}+b^{2} }}\\\\\ P(k,1)\\\\\ r:x-y=2\\\ r:x-y-2=0\\\\\ d= \sqrt{2} \\\\\\\ \dfrac{|x-y-2|}{ \sqrt{a^{2}+b^{2} }}= \sqrt{2} \\\\\\\ \dfrac{|k-1-2|}{ \sqrt{1^{2}+(-1)^{2} }}= \sqrt{2} \\\\\\\ \dfrac{|k-3|}{ \sqrt{1+1 }}= \sqrt{2} \\\\\\\ \dfrac{|k-3|}{ \sqrt{2 }}= \sqrt{2} \\\\\\\ |k-3|= \sqrt{2} \times \sqrt{2}\\\\\\ |k-3|= \sqrt{4} \\\\\\ k-3=2 \\\\\\ k=2+3\\\\ \boxed{k=5}$

$P(5,1)$

Boa tarde!
Bons estudos!