find all real numbers x so that the point (0,2) is exactly 5 units from (x,2x)

Question

I understand the pythegorean therum but do not know how to solve with the 2x

Answer 1

All points (x,2x) are represented by the line y=2x (green).

All points equidistant (5 units) from (0,2) are represented by the circle x²+(y-2)²=25 (blue).

The line intersects the circle when y=2x:

x²+(2x-2)²=25, x²+4x²-8x+4=25, 5x²-8x-21=0=(5x+7)(x-3), that is, x=-7/5=-1.4 and 3.

The real values are -1.4 and 3. The points A(-7/5,-14/5) and B(3,6) are radial, where the circle’s centre is (0,2).