# Distance Between Two Parallel Lines

We know that the perpendicular distance from origin to a straight line whose equation is a+by+c=0 is |c|/√(a^{2}+b^{2}). Also, we know that the perpendicular distance from (h,k) to ax+by+c=0 is, |ah+bk+c|/√(a^{2}+b^{2}). These two are the formulas to find out the distances to a line from origin or a point. But, how to find the distances between two parallel lines? Let us derive the formula for the distance between two parallel lines.

The general form of equation of a straight line is ax+by+c=0 whose slope is –a/b.

We know the fact that any two lines are parallel if they have same slope. Thus, any two parallel lines have same coefficients of x and y but differs in constants.

Let ax+by + p =0 and ax+by + q =0 be two parallel lines and let us find the distance between these lines now.

Let (h,k) be a point of the line ax+by+p=0

Then ah+bk +p=0 à ah+bk =-p …(1)

Now distance from (h,k) to the second line ax+by +q=0 according to formula is,

|ah+bk+q|/√(a^{2}+b^{2})

=|-p+q |/√(a^{2}+b^{2}) (By using equation (1))

=|p-q |/√(a^{2}+b^{2})

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