SOLUTION: Find the two equations of the lines parallel to the line x+2y-5 = 0 and passing at a distance 2 from the origin.

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Question 1080804: Find the two equations of the lines parallel to the line x+2y-5 = 0 and passing at a distance 2 from the origin.
Found 2 solutions by ikleyn, josgarithmetic:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
The general equation of a straight line parallel to the given line is

x + 2y + c = 0,

where "c" is some constant.


If we want our straight line be parallel to the given line and pass at the distance of 2 from the origin (x,y) = (0,0), 
we must find "c" from this equation


abs%28c%29%2Fsqrt%281%5E2+%2B+2%5E2%29 = 2    (1)


(see the lesson The distance from a point to a straight line in a coordinate plane in this site).


From equation (1), |c| = 2%2Asqrt%285%29,  so c can have two values: c = 2%2Asqrt%285%29  or  c = -2%2Asqrt%285%29.


Thus, the two equations the problem asks for are

x+%2B+2y+%2B+2%2Asqrt%285%29 = 0   (2)

and

x+%2B+2y+-+2%2Asqrt%285%29 = 0   (3).

Solved.


Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"The distance from a point to a straight line in a coordinate plane".



Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
y=2x, perpendicular line going through Origin
Points (x, 2x).

%28x-0%29%5E2%2B%282x-0%29%5E2=2%5E2
x%5E2%2B4x%5E2=4
5x%5E2=4
x=0%2B-+sqrt%28%284%2F5%29%29
x=0%2B-+2%2Fsqrt%285%29

y=-4%2Fsqrt%285%29
y=-4%2Asqrt%285%29%2F5
OR
y=4%2Asqrt%285%29%2F5

Line wanted
slope must be -1%2F2 and either contain (-2*sqrt(5)/5, -4*sqrt(5)/5 ) or ( 2*sqrt(5)/5, 4*sqrt(5)/5 ).
ONE of the lines: highlight_green%28y-4%2Asqrt%285%29%2F5=-%281%2F2%29%28x-2%2Asqrt%285%29%2F5%29%29; you can simplify this, and form the other line...