The gradient (slope) of a line is the tangent of the angle, the line makes with the x-axis. And . If the gradient is m then or, equivalently, .
And, of course, if length of the line is d, then so that and .
Putting that into the equation for the gradient, m, .
Square both sides to get which is the same as .
Solve that for and then find from
and are the changes in the x and y coordinates to add then to the x and y values of the initial point to find the final point.
Yes, there are two possible answers- the points on the same line on either side of the initial point. When you solve , you will need to take a square root and which point you get depends on which sign you take for .