If the given circle is then we may complete the square on the variables to obtain the standard form:

Now, for some point outside the circle, we may form a right triangle by drawing 3 line segments:

One from the center of the circle to the tangent point whose length is equal to the radius of the circle, one from the tangent point to , and one from the center of the circle to . So, by Pythagoras, we find:

Now, equating this to the distance between and we find:

Expanding and simplifying, we will find the required locus.