Question

6. Qual é a equação da reta que é paralela à reta de equação 3x – 5y + 10 = 0 e que passa pelo ponto M(2, 4)?

Answer 1

$\boxed{\begin{array}{l}\rm Considere~ duas~retas\\\rm\ell_1:m_1x+n_1\\\rm\ell_2:m_2x+n_2\\\underline{\sf ent\tilde ao\!:}\\\rm m_1=m_2~e~n_1\ne n_2\implies\,\ell_1~e~\ell_2~paralelas\\\rm m_1=m_2~e~n_1=n_2\implies\,\ell_1~e~\ell_2~coincidem\\\rm m_1\ne m_2\implies\,\ell_1~e~\ell_2~concorrentes\\\rm m_1\cdot m_2=-1\implies\,\ell_1~e~\ell_2~perpendiculares\end{array}}$

$\large\boxed{\begin{array}{l}\rm r:3x-5y+10=0\longrightarrow m_r=-\dfrac{3}{-5}=\dfrac{3}{5}\\\rm s: m_sx+by+c=0\\\rm m_r\parallel m_s\iff m_r=m_s\\\\\rm m_s=\dfrac{3}{5}\\\rm y=y_0+m_s\cdot (x-x_0)\\\rm y=4+\dfrac{3}{5}\cdot(x-2)\\\rm y=4+\dfrac{3}{5}x-\dfrac{6}{5}\bullet(\bf5)\\\rm 5y=20+3x-6\\\rm 3x-5y+14=0\end{array}}$