Question 639450
Find the exact directed distance from the line 3x-4y=12 to the point (-2,-1).
<pre>

{{{drawing(400,400,-5,5,-6,4, line(-6,-7.5,6,1.5), graph(400,400,-5,5,-6,4), green(line(-2,-1,-.32,-3.24)), locate(-2,-1,"(-2,-1)"), circle(-2,-1,.1),
locate(-1.5,-2,d)


  )}}}




Use the formula:

The distance from the line Ax+By+C=0 to the point (h,k) is given
by the formula

d = ±{{{abs((Ah+Bk+C)/sqrt(A^2+B^2)))}}} where the sign is the sign
of {{{B/A}}} 

For your equation

3x-4y=12, first get 0 on the right:

3x-4y-12=0

Then compare to

Ax+By+C=0

and get A=3, B=-4, C=-12, and (h,k) = (-2,-1), so h=-2 and k=-1

{{{B/A}}} = {{{(-4)/3}}} = {{{-4/3}}} which is negative, so the
directed distance will be negative:

Substituting:

d = ±{{{abs((Ah+Bk+C)/sqrt(A^2+B^2))}}}

d = -{{{abs(((3)(-2)+(-4)(-1)+(-12))/sqrt((3)^2+(-4)^2))}}}

d = -{{{abs((-6+4-12)/sqrt(9+16))}}}

d = -{{{abs((-14)/sqrt(25))}}}

d = -{{{abs((-14)/5)}}}

d = -{{{14/5}}} = -2.8

And that green line looks a little shorter than 3 units long.
That's only a "ballpark" check. But if you compare it to 3
units along the x-axis or the y-axis, you'll agree that 2.8
is at least close to the length of that green line.

Edwin</pre>