Question

A reta determinada pelos pontos a(2, 3) b(k, 3k-1) tem inclinação 45° em relação ao eixo das abcissas. Calcule k

Answer 1

✅ Após resolver os cálculos, concluímos que o valor do parâmetro "k" é:

                  $\Large\displaystyle\begin{gathered} k = 1\end{gathered}$

Sejam os dados:

               $\Large\begin{cases} A(2, 3)\\B(k,\,3k - 1)\\\theta = 45^{\circ}\\k = \:?\end{cases}$

Sabemos que a inclinação da reta é o ângulo que a referida reta forma com o eixo das abscissas no seu sentido positivo e para calcular o seu valor devemos utilizar a seguinte fórmula:

$\Large\displaystyle\begin{gathered} \bf I\end{gathered}$            $\Large\displaystyle\begin{gathered} \theta = \arctan(m_{r})\end{gathered}$

Desenvolvendo a equação "I", temos:

             $\Large\displaystyle\begin{gathered} \theta = \arctan(m_{r})\end{gathered}$

             $\Large\displaystyle\begin{gathered} \theta = \arctan(\tan\theta)\end{gathered}$

             $\Large\displaystyle\begin{gathered} \theta = \arctan\bigg(\frac{\sin\theta}{\cos\theta}\bigg)\end{gathered}$

$\Large\displaystyle\begin{gathered} \bf II\end{gathered}$         $\Large\displaystyle\text{ \begin{gathered} \theta = \arctan\bigg(\frac{y_{B} - y_{A}}{x_{B} - x_{A}}\bigg)\end{gathered} }$

Sabemos que:

                       $\Large\displaystyle\begin{gathered} \tan45^{\circ} = 1\end{gathered}$

Então:

$\Large\displaystyle\begin{gathered} \bf III\end{gathered}$           $\Large\displaystyle\text{ \begin{gathered} \frac{y_{B} - y_{A}}{x_{B} - x_{A}} = \tan45^{\circ}\end{gathered} }$

Substituindo os dados, bem como a tangente de 45° na equação "III", temos:

                   $\Large\displaystyle\begin{gathered} \frac{3k - 1 - 3}{k - 2} = 1\end{gathered}$

                            $\Large\displaystyle\begin{gathered} \frac{3k - 4}{k - 2} = 1\end{gathered}$

                             $\Large\displaystyle\begin{gathered} 3k - 4 = k - 2\end{gathered}$

                             $\Large\displaystyle\begin{gathered} 3k - k = -2 + 4\end{gathered}$

                                      $\Large\displaystyle\begin{gathered} 2k = 2\end{gathered}$

                                         $\Large\displaystyle\begin{gathered} k = \frac{2}{2}\end{gathered}$

                                         $\Large\displaystyle\begin{gathered} k = 1\end{gathered}$

Portanto, o valor do parâmetro "k" é:

                                         $\Large\displaystyle\begin{gathered} k = 1\end{gathered}$

Neste caso, os pontos são:

                                       $\Large\begin{cases} A(2, 3)\\B(1, 2)\end{cases}$

$\Large\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Prova:\:\:\:}}}\end{gathered} }$

            $\Large\displaystyle\text{ \begin{gathered} \theta = \arctan\bigg(\frac{y_{B} - y_{A}}{x_{B} - x_{A}}\bigg)\end{gathered} }$

       $\Large\displaystyle\begin{gathered} 45^{\circ} = \arctan\bigg(\frac{3\cdot1 - 1 - 3}{1 - 2}\bigg)\end{gathered}$

       $\Large\displaystyle\begin{gathered} 45^{\circ} = \arctan\bigg(\frac{3 - 4}{1 - 2}\bigg)\end{gathered}$

        $\Large\displaystyle\begin{gathered} 45^{\circ} = \arctan\bigg(\frac{-1}{-1}\bigg)\end{gathered}$

        $\Large\displaystyle\begin{gathered} 45^{\circ} = \arctan(1)\end{gathered}$

        $\Large\displaystyle\begin{gathered} 45^{\circ} = 45^{\circ}\end{gathered}$

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

Saiba mais:

1. https://brainly.com.br/tarefa/24007775
2. https://brainly.com.br/tarefa/13153342
3. https://brainly.com.br/tarefa/22403498
4. https://brainly.com.br/tarefa/921473
5. https://brainly.com.br/tarefa/44509586
6. https://brainly.com.br/tarefa/32816745
7. https://brainly.com.br/tarefa/40747897
8. https://brainly.com.br/tarefa/33104372
9. https://brainly.com.br/tarefa/25491927
10. https://brainly.com.br/tarefa/4970489

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