Question

Calcule a distância entre o ponto P e a reta r em cada caso:

a) P(1,2) e (r) 2x-3y+1=0 b) P(-2,-5) e (r) x+2y-3=0 c) P(4,-1) e (r) x-y=0
d) P(-1,2) e (r) 3x-5=0 e) P(-3,-2) e (r) 2y+1=0

Answer 1

a)

$D_{p, r}=\frac{|a. x_{p}+b. y_{p}+c|}{\sqrt{{a}^{2}+{b}^{2}}}$

$D_{p, r}=\frac{|2. 1-3.2+1|}{\sqrt{{2}^{2}+{(-3)}^{2}}}$

$D_{p, r}=\frac{|2-6+1|}{\sqrt{4+9}}$

$D_{p, r}=\frac{|-3|}{\sqrt{13}}$

$D_{p, r}=\frac{3}{\sqrt{13}}$

$\boxed{\boxed{D_{p, r}=\frac{3\sqrt{13}}{13}}}$

b)

$D_{p, r}=\frac{|a. x_{p}+b. y_{p}+c|}{\sqrt{{a}^{2}+{b}^{2}}}$

$D_{p, r}=\frac{|1.(-2)+2.(-5)-3|}{\sqrt{{1}^{2}+{2}^{2}}}$

$D_{p, r}=\frac{|-2-10-3|}{\sqrt{1+4}}$

$D_{p, r}=\frac{|-15|}{\sqrt{5}}$

$D_{p, r}=\frac{15}{\sqrt{5}}=\frac{15\sqrt{5}}{5}$

$\boxed{\boxed{D_{p, r}=3\sqrt{5}}}$

c) P(4,-1) e (r) x-y=0

$D_{p, r}=\frac{|a. x_{p}+b. y_{p}+c|}{\sqrt{{a}^{2}+{b}^{2}}}$

$D_{p, r}=\frac{|1.4-1.(-1)|}{\sqrt{{4}^{2}+{(-1)}^{2}}}$

$D_{p, r}=\frac{|4+1|}{\sqrt{16+1}}}$

$D_{p, r}=\frac{5}{\sqrt{17}}=\frac{5\sqrt{17}}{17}$

$\boxed{\boxed{D_{p, r}=\frac{5\sqrt{17}}{17}}}$