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What if

\\ {\frac{a x_{1}+b y_{1}+c z_{1}+d}{a x_{2}+b y_{2}+c z_{2}+d}=0} \;\;\;\;\;\;\;\;\;\;\;\; {\text { (same side) }} \\\\ {\frac{a x_{1}+b y_{1}+c z_{1}+d}{a x_{2}+b y_{2}+c z_{2}+d}=\infty} \;\;\;\;\;\;\;\;\;\;\;{\text { (opposite side) }}
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In that case, one of the points lies in the plane
if the value is zero then quotient point is in the plane and vice versa

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