Question

As retas 2x-5y-3=0 e (p+1)x-y+11=0 são paralelas, encontre o valor de p. Marque a alternativa que corresponde ao produto de p com o coeficiente angular das retas.

Answer 1

Posição relativa de duas retas

Considere duas retas

$\mathsf{\ell_{1}:m_{1}x+n_{1}}\\\mathsf{\ell_{2}:m_{2}x+n_{2}}$.

Então

$\mathsf{m_{1}=m_{2}~e~n_{1}\ne n_{2}\implies\,\ell_{1}~e~\ell_{2}~paralelas}\\\mathsf{m_{1}=m_{2}~e~n_{1}=n_{2}\implies\,\ell_{1}~e~\ell_{2}~coincidem}\\\mathsf{m_{1}\ne m_{2}\implies\,\ell_{1}~e~\ell_{2}~concorrentes}\\\mathsf{m_{1}\cdot m_{2}=-1\implies\,\ell_{1}~e~\ell_{2}~perpendiculares}$

$\dotfill$

$\mathsf{r:2x-5y-3=0\implies~m_r=\dfrac{2}{5}}\\\mathsf{s:(p+1)x-y+11=0\implies~m_s=p+1}\\\mathsf{m_s=m_r}\\\mathsf{p+1=\dfrac{2}{5}}\\\mathsf{5p+5=2}\\\mathsf{5p=2-5}\\\mathsf{5p=-3}\\\mathsf{p=-\dfrac{3}{5}}\\\mathsf{p\cdot\dfrac{2}{5}=\left(-\dfrac{3}{5}\right)\cdot\dfrac{2}{5}=-\dfrac{6}{25}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{\maltese~~alternativa~~c}}}}}$

$\dotfill$

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