SOLUTION: 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

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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) About Me  (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 y-1=-%28x-1%29,
y=-x%2B1%2B1
y=-x%2B2

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(52800) About Me  (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  sqrt%282%29  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)