Question 1011469: Find the equation of the locus of a point P (x,y) that follows the condition of:
P is sqroot (2) units from the line with equation y = x Found 2 solutions by josgarithmetic, ikleyn:Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website! They would be either or both of two parallel lines to y=x.
Pick any convenient point on the line y=x, and find the line perpendicular to y=x at that point; find the equation for this perpendicular line. Try point (1,1) on y=x. The perpendicular line is ,
This perpendicular line is a general point (x, -x+2).
You want distance from (x, -x+2) to (1,1) to be square-root of 2.
Use the distance formula.
Do you find a pathway to continue and finish? Answer by ikleyn(52803) (Show Source): You can put this solution on YOUR website! .
This locus consists of two straight lines parallel to the given line y = x (shown in black in the Figure below)
and remoted from the given line at the distance in the normal direction.
It means that one of these two straight lines passes through the point (-1,1), while the other straight line
passes through the point (1,-1).
Having this, you can easily understand that the first straight line has the equation y = x+2.
It is shown in red color in the Figure.
The second straight line has the equation y = x-2. It is shown in green color in the Figure.
If you want to have one equation describing both these lines, here it is:
|y - x| = 2.
Figure. y=x (black); y=x+2 (red) and y=x-2 (green)