Question

calcule a distância entre o ponto p e a reta r nos casos:

P (-4,-7) e r:3x+4y-20=0
P (4,2)e r:6x-8y-13=0

Answer 1

Equação da distância entre um ponto e uma reta é $\mathsf{D=\dfrac{ax+by+c}{\sqrt{(a^2+b^2)}}}$

Substituindo os dados na equação temos:

$\mathsf{P (-4,-7)}\\\mathsf{r:3x+4y-20=0}\\\\\mathsf{D=\dfrac{3.(-4)+4.(-7)-20}{\sqrt{(3^2+4^2)}}}\\\\\\\\\mathsf{D=\dfrac{-12-28-20}{\sqrt{9+16}}}\\\\\\\mathsf{D=\Big|\dfrac{-60}{\sqrt{25}}\Big|}\\\\\\\mathsf{D=\dfrac{60}{5}}\\\\\\\boxed{\mathsf{D=12~u.m}}$

$\mathsf{P (4,2)}\\\mathsf{r:6x-8y-13=0}\\\\\\\mathsf{D=\dfrac{6.4-8.2-13}{\sqrt{(4^2+2^2)}}}\\\\\\\mathsf{D=\dfrac{24-16-13}{\sqrt{16+4}}}\\\\\mathsf{D=\Big|\dfrac{-5}{\sqrt{20}}\Big|}\\\\\\\mathsf{D=\dfrac{5}{\sqrt{20}}.\dfrac{\sqrt{20}}{\sqrt{20}}}}\\\\\\\mathsf{D=\dfrac{\diagup\!\!\!\!5\sqrt{20}}{\diagup\!\!\!\!\!20}}\\\\\\\boxed{\mathsf{D=\dfrac{\sqrt{20}}{4}~u.m}}$



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