What is the equation of the locus for y=2x and y=-2x, and what is the equation of the locus for y=2 and y=8? These are both equidistant.
(1) The locus of points equidistant from the lines `y=2x` and `y=-2x` .
Note that these two lines intersect at the origin.
The locus of points equidistant fromthe two lines are the points on the line x=0 (the y-axis) and the points on the line y=0 (the x-axis).
You can see this from the graphs.
Another way to realize this is to see that y=-2x is a reflection of the line y=2x about the x-axis [If g(x)=-2x and f(x)=2x then g(x)=-f(x)] and also a reflection over the y-axis [g(x)=f(-x)]
Note that the lines form vertical angles. The axes bisect the angles and from geometry we have a point on the angle bisector is equidistant from the sides of the angle.
(2) The locus of points equidistant to the lines `y=2"and"y=8` is the line y=5
Each point on y=5 is three units from y=2 and y=8.