So the function you are dealing with is a circle of radius centered at . This may be a helpful picture to have in your mind while doing this.
Given that , the horizontal tangent occurs when the numerator is zero (because then the derivative is 0), which is at . Use the function to find that .
Similarly, the vertical tangent occurs when the denominator is zero (because then the derivative is "infinity"), which is at . Use the function to find that .
So the horizontal tangent lines occur at and and the vertical tangent lines occur at and .
If you think about the picture of the circle, this makes a lot of sense.