Derivation of the Formula to Calculate the Distance Between Two Parallel Lines
Formulas
$$ d=\frac { |2k| }{ \sqrt { m^{ 2 }+1 } } $$
Explanation
When solving problems involving the tangent to a conic section, one often needs to calculate the distance between two tangents. While it’s not particularly challenging, thanks to the formula for the distance from a given point to a line, having an easy and quick formula for this distance can help to reduce calculation time.
Derivation
Let’s assume two parallel lines have the equation $y=mx\pm k$. The distance from any point $(x,y)$ to the line $y=mx+k$ is $$ \frac { |mx-y+k| }{ \sqrt { m^{ 2 }+1 } } $$ For a point $(x_1,y_1)$ on the line $y=mx-k$, we have $$ k=mx_1-y_1 $$ Substituting $mx_1-y_1=k$ into the distance formula, we get $$ {{ |mx_{1}-y_{1}+k| }\over{ \sqrt { m^{ 2 }+1 } }} = {{ |k+k| }\over{\sqrt { m^{ 2 }+1 }}} $$ Therefore, the distance between the two parallel lines $y=mx\pm k$ is $$ \frac { |2k| }{ \sqrt { m^{ 2 }+1 } } $$
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