Q.Find the value(s) of $p$ if the distance of the point $(2, p)$ from the line $3x - 4y - 5 = 0$ is $4$ units.

[!TLDR] Set $\dfrac{|6-4p-5|}{5}=4$ and solve the resulting modulus equation for $p$. [!ANSWER] …

$d=\dfrac{|3(2)-4p-5|}{\sqrt{3^2+(-4)^2}}=\dfrac{|1-4p|}{5}=4 \implies |1-4p|=20$. Case (+): $1-4p=20 \implies -4p=19 \implies p=-\dfrac{19}{4}$. Case (−): $1-4p=-20 \implies -4p=-21 \implies p=\dfrac{21}{4}$. Both are genuine solutions, since a point …

Substitute the point (with unknown $p$) into the distance formula, set it equal to the given distance, and solve the resu …

Dropping one of the two cases of the modulus equation, or making a sign error while simplifying $3 …