Let $l$ be a line described by equation ax+by+c=0 and let $P(x,y)$ be a point not on $l$ . Express the distance, $d$ between $l and P$ in terms of the coefficients $a, b and c$ of the equation of line?

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Answer 1 · 2016-11-14T19:04:58

$d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)$

Let $l->a x + b y + c=0$ and $p_0 = (x_0,y_0)$ a point not on $l$ .

Supposing that $b ne 0$ and calling $d^2=(x-x_0)^2+(y-y_0)^2$ after substituting $y=-(a x+c)/b$ into $d^2$ we have

$d^2=(x - x_0)^2 + ((c + a x)/b + y_0)^2$ . The next step is find the $d^2$ minimum regarding $x$ so we will find $x$ such that

$d/(dx)(d^2) = 2 (x - x_0) - (2 a ((c + a x)/b + y_0))/b = 0$ . This occours for

$x = (b^2 x_0 - a b y_0-a c)/(a^2 + b^2)$ Now, substituting this value into $d^2$ we obtain

$d^2=(c + a x_0 + b y_0)^2/(a^2 + b^2)$ so

$d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)$

Let ll be a line described by equation ax+by+c=0 and let P(x,y)P(x,y) be a point not on ll. Express the distance, dd between l and Pl and P in terms of the coefficients a, b and ca, b and c of the equation of line?