Since x is a variable, and the center of a circle is a constant, to
avoid a conflict of letters, I will change the problem so that the
center is (c,3c) and not (x,3x). So consider the problem as:
the center of a circle is (c,3c). Find c if the circle passes through
the origin and the diameter is 10 units.
The equation of any circle is
where the center is (h,k) and the radius is r.
So (h,k) = (c,3c) and r=5 units (because the diameter
is 10 units and the radius is one-half the diameter).
Plugging in:
Since it passes through the origin, we can
substitute (x,y) = (0,0). Plugging in:
There are two answers. Notice that both circles pass through
the origin and both are 10 units across, that is, the diameter
is 10 units, as you can see by the green diameters below:
So there are two solutions,
and 
You should point out to your teacher that there is a conflict
of letters. x cannot represent a variable if it is supposed
to represent a constant.
Edwin
