Question

determine a distancia do ponto p (2 1) a reta 4x +3y +13

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo a passo:

$\mathsf{r:4x + 3y + 13 = 0}$

$\mathsf{P(2;1)}$

$\mathsf{d_{p,r} = |\:\dfrac{ax_0 + by_0 + c}{\sqrt{a^2 + b^2}}\:|}$

$\mathsf{d_{p,r} = |\:\dfrac{4.2 + 3.1 + 13}{\sqrt{4^2 + 3^2}}\:|}$

$\mathsf{d_{p,r} = |\:\dfrac{8 + 3 + 13}{\sqrt{16 + 9}}\:|}$

$\mathsf{d_{p,r} = |\:\dfrac{24}{\sqrt{25}}\:|}$

$\boxed{\boxed{\mathsf{d_{p,r} = \dfrac{24}{5}}}}$

Answer 2

✅ Após resolver os cálculos, concluímos que a distância entre o ponto "P" e a reta "r" é:

                  $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\:\bf d_{\overline{Pr}} = \frac{24}{5}\:\:\:}}\end{gathered} }$

Sejam os dados:

             $\Large\begin{cases} P(2, 1)\\r: 4x + 3y + 13 = 0\end{cases}$

Para calcular a distância entre o ponto "P" e a reta "r", devemos utilizar a seguinte fórmula:

$\Large\displaystyle\begin{gathered} \bf(I)\end{gathered}$        $\Large\displaystyle\text{ \begin{gathered} d_{\overline{Pr}} = \frac{|a_{r}x_{P} + b_{r}y_{P} + c_{r}|}{\sqrt{a_{r}^{2} + b_{r}^{2}}}\end{gathered} }$

Substituindo os valores na equação "I", temos:

       $\Large\displaystyle\text{ \begin{gathered} d_{\overline{Pr}} = \frac{|4\cdot2 + 3\cdot1 + 13|}{\sqrt{4^{2} + 3^{2}}}\end{gathered} }$

                 $\Large\displaystyle\text{ \begin{gathered} = \frac{|8 + 3 + 13|}{\sqrt{16 + 9}}\end{gathered} }$

                  $\Large\displaystyle\text{ \begin{gathered} = \frac{|24|}{\sqrt{25}}\end{gathered} }$

                  $\Large\displaystyle\begin{gathered} = \frac{24}{5}\end{gathered}$

✅ Portanto, a distância procurada é:

          $\Large\displaystyle\text{ \begin{gathered} d_{\overline{Pr}} = \frac{24}{5}\end{gathered} }$

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Observe \:o\:Gr\acute{a}fico!!\:\:\:}}}\end{gathered} }$