Help with lines and distance?

Question

In this question you will derive a general formula for the distance of a point from a line.

Let P be the point (p,q) and L the line y=mx+b.

The slope of L is _____(which I know is m)

The slope of a line perpendicular to L is __________(which i know is -(1/m).

The line through P perpendicular to L can be written as y=sx+c

where s is: ________(which i know is -(1/m)

and c is:_______

That line intersects L in the point Q = (u/v)

where u is: _______

and v is: __________

The distance of P and Q is ______________.

All answers must be in terms of m,b, p, and q.

Answer 1

If the perpendicular line goes through P, then

q = (-1/m)p + c

so c = q + (1/m)p

the point Q satisfies two equation:

v = mu + b, and

v = su + c

so you have two equations for the two unknowns, u and v.

To solve for u:

mu + b = (-1/m)u + q + (1/m)p so

u(m + 1/m) = q + (1/m)p - b

etc.

To solve for v directly

u = (v - b)/m = (v - c)/s so

v/m - v/s = b/m - c/s

v(1/m + m) = b/m - q - (1/m)p = (b - p)/m - q

etc.