Question

How do I find the distance from (6,1) to the line defined by y=3x-7. Express as a radical or a number rounded to the nearest hundredth.

Answer 1

This is what I would do, but there might be an easier way: find the line perpendicular to y=3x-7 that goes through (6,1). Than find the intersection, (a,b), of those two lines. The distance then will be sqrt((6-a)^2+(1-b)^2)

Answer 2

\[d=|\sqrt{Ax1+By1+C}|/\sqrt{A^2+B^2}\]

Answer 3

Now the shortest distance is when Lines are parallel.

y = 3x - C, passes through (6,1)
1 = 18 -C
C=17

Equation becomes y = 3x -17
Now there is a formula to find distance between parallel lines.
\[d = \frac{|C_1 -C_2|}{\sqrt{a^2 +b^2}}\]

ax + by + c = 0 is the standard form.
line equation is standard form y -3x +7 = 0 and y -3x -17 = 0
a = -3
b = 1
\[C_1 = 7\]
\[C_2 = -17\]
\[d = \frac{17 + 7 }{\sqrt{9 +1}}\]
\[d = \frac{24}{\sqrt{10}}\]