I already found one solution, I'm just curious if anybody else can find another.

A circle goes through B (-1,0) and is tangent to a line y=2x at A (1,2). What is the surface of this circle?

Surface? You mean area?
Basically you want to find the center of the circle though. You can use the fact that a tangent to a circle is perpendicular to a line from the center of the circle to the tangent point. Recall perpendicular refers to a negatove reciprocal slope. Use this fact to construct two equations and plug in the points to find center (x,y) of the circle.

This post was edited by Zekdawg on May 28 2017 09:03am

Haha, surface/area got lost in translation there. What you mention now is what I did to solve this one. Good find, though. Any other possibility?

Due to the nature of the problem, I do not believe that there is more than one solution due to the fact that you are dealing with a fixed tangent point (not tangent line) and a point the circle must go through precisely. With any other radius, intersecting (1,2) and being tangent to the line at (-1,0) is not possible. Imagine that the circle starts off small from the tangent point and then grows until the other point is reached (solution). There is only one radius that this occurs at, the rest the point (-1,0) is either inside or outside of the circle.

If it simply had to be tangent to the line, then there would be infinite possibilities with infinite radii, but then the question could not point you toward an objective answer. It restricts the problem in this way so that there is only one objective answer.

This post was edited by Dontrunaway on May 29 2017 02:26pm

Indeed, that makes sense. How silly that I didn't think of that.
Well, learn something new every day. I can only know so much when proceeding in my crash course