Question

Para que valores de h, as retas r: 6x-4y+10=0 e hx-y+1=0 são concorrentes?

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

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

$\dotfill$

$\mathsf{r:6x-4y+10=0~m_{1}=\dfrac{3}{2}}\\\mathsf{s:hx-y+1=0\implies m_2=h}\\\mathsf{h\ne\dfrac{3}{2}\implies~r\times s}$

$\dotfill$

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