Answer
$(x+\sqrt 2)^2-(y+\sqrt 2)^2=2$
Work Step by Step
Step 1. If the circle is tangent to both axes, the absolute values of the center coordinates $(a,b)$ should both equal to the radius, that is $|a|=|b|=\sqrt 2$
Step 2. If the circle is in the third quadrant, we have $a=b=-\sqrt 2$
Step 3. The equation can be found as $(x+\sqrt 2)^2-(y+\sqrt 2)^2=2$
