How to determine the position of a circle on the coordinate plane?

Question

Geometry

We have a coordinate plane and a circle.

Input data: X, Y, R - circle radius.

Using these data, you need to determine how many quarters of the coordinate plane the circle touches (the x, y coordinates are the center of the circle and the coordinate axis is also considered).

How?

Answer 1

Consider the first quarter. Write down the geometric condition when the circle intersects (touches?). Initially, the counter is 0. If the condition for the first is met, then increase it by 1. Similarly for the other three

Answer 2

Consider options. What does it mean that the circle touches one quarter? This means that it does not intersect with any of the coordinate axes, i.e. the lowest point of the circle is above OX, or the highest point is below OX and similarly for OY.
Can a circle touch two quarters? Easy. If we exclude the first case first, then it turns out that the circle must intersect exactly one of the axes. Further, can a circle touch three-quarters? Draw, think, and understand that it is not (hint - the circle is a convex figure, the segment between any two points in the circle lies entirely in the circle). Only the fourth option remains. Those. if not 1 or 2 quarters, then exactly 4.