Linear function (difficult for me, SAT Math II level)

[rainyrabbit]

If point P(m,2m) is 5 units from the line 12x+5y=1, m could equal
a. 43/11
b.-3
c.-65/22
d.5
e.3

The answer is E.


Explanation given in book:
- distance = abs(12(m) + 5(2m) -1) / root(144+25) = 5.
>>> m=3 or -32/11

I do not understand this explanation (of course finding m from the equation provided in the explanation is no problem, but I don't know how this equation was set up). Could you please help me out?

 

[ZioX]

root(144+25)=13

solve for m.

 

[uart]

I do not understand this explanation (of course finding m from the equation provided in the explanation is no problem, but I don't know how this equation was set up). Could you please help me out?

It uses the following well know a formula for the perpendicular distance from a point to a line.

The distance "d" from the point (x1,y1) to the line ax+by+c=0 is given by :

d = |a x1 + b y1 + c| / sqrt(a^2 + b^2)

 

[uart]

BTW there are quite a few differnet methods you can use to come up with a formula for the distance from a point to a line. The one quoted above is popular but I prefer the following because it has a simple one line proof based on similar triangles.

The distance from a given point to the line y=mx + b is given by,

d = v/sqrt(m^2+1), where v is the vertical distance from the point to the line.

 

[rainyrabbit]

Thank you very much, uart! I greatly appreciate your help.