Question

05- Considerando que a distância entre ponto P(0, K) e a reta r, de equação 4x + 3y - 2 = 0, é igual a 4 unidades, calcule o valor da coordenada k.

Answer 1

$\displaystyle \text{reta}:\text {a.x + b.x + c}=0 \ \ ; \ \text {P}(\text x_\text o,\text y_\text o) \\\\ \text{Dist{\^a}ncia do ponto {\`a} reta}: \\\\ \text D = \frac{|\text a.\text x_\text o+\text b.\text y_\text o+\text c|}{\sqrt{\text a^2+\text b^2}} \\\\\\ \text{temos}: \\\\ \text{reta}: 4\text x+3\text y-2=0 \ \ ; \ \text P(0,\text k) \ ; \ \text D = 4$

substituindo os respectivos valores na distância do ponto à reta :

$\displaystyle 4=\frac{|4.0+3.\text k-2|}{\sqrt{4^2+3^2}} \to 4=\frac{|3\text k-2|}{\sqrt{16+9}} \\\\\\ 4 = \frac{|3\text k-2|}{5} \to |3\text k-2| = 20$

Aplicando a definição de módulo :

$\displaystyle 3\text k -2 = 20 \to 3\text k = 22 \to \boxed{\text k = \frac{22}{3}\ }\checkmark \\\\\\ 3\text k-2 = -20 \to 3\text k = -18 \to \boxed{\text k = -6\ }\checkmark$