SOLUTION: Which of the lines 2x-y+3=0 and x-4y-7=0 is farther from origin?

Algebra ->  Length-and-distance -> SOLUTION: Which of the lines 2x-y+3=0 and x-4y-7=0 is farther from origin?      Log On


   

Question 1043689: Which of the lines 2x-y+3=0 and x-4y-7=0 is farther from origin?
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Find the slope of each line.
2x-y%2B3=0
y%5B1%5D=2x%2B3
m%5B1%5D=2
.
.
.
x-4y-7=0
4y=x-7
y%5B2%5D=x%2F4-7%2F4
m%5B2%5D=1%2F4
.
.
.
The perpendicular line from the origin to each line has a slope that is the negative reciprocal of the given slope.
m%5B1p%5D=-1%2F2
m%5B2p%5D=-4
So then each line goes through the origin so the equation of each line is,
y-0=-1%2F2%28x-0%29
y%5B1p%5D=-%281%2F2%29x
.
.
.
y-0=-4%28x-0%29
y%5B2p%5D=-4x
.
.
.
Find the intersection point of each line,
2x%2B3=-%281%2F2%29x
%285%2F2%29x=-3
x=-6%2F5
So then,
y=-%281%2F2%29%286%2F5%29
y=3%2F5
(-6%2F5,3%2F5)
and
x%2F4-7%2F4=-4x
%2817%2F4%29x=7%2F4
x=7%2F17
Then,
y=-4%287%2F17%29
y=-28%2F17
(7%2F17,-28%2F17)
So then calculate the distance from the origin to the respective intersection points,
D%5B1%5D%5E2=%28-6%2F5-0%29%5E2%2B%283%2F5-0%29%5E2
D%5B1%5D%5E2=%2836%2F25%29%2B%289%2F25%29
D%5B1%5D%5E2=45%2F25
D%5B1%5D=sqrt%2845%29%2F5
D%5B1%5D=%283%2F5%29sqrt%285%29
.
.
.
D%5B2%5D%5E2=%287%2F17-0%29%5E2%2B%28-28%2F17-0%29%5E2
D%5B2%5D%5E2=%2849%2B784%29%2F17%5E2
D%5B2%5D=sqrt%28833%29%2F17
.
.
.
Here's a graphical representation of the problem.
.
.
.
.

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
.
Which of the lines 2x-y+3=0 and x-4y-7=0 is farther from origin?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To answer your question, you have to calculate the distance from the point (0,0) (the origin of a coordinate system) to the given lines.

Read the lesson
    - The distance from a point to a straight line in a coordinate plane
in this site on how to do it.