Another purportedly hard geometry puzzle

One can rotate Figure 1 anticlockwise by 90° and flip it horizontally without changing the radius of the smaller circle. The hope is that this will make the problem easier to solve. This is Figure 3 below. <img alt="A circle is nestled between three curves on the positive x y plane. First, the concave edge of a circle of radius one centred at the origin. Second, the convex edge of the curve y equals x squared. Third, the x axis. What is the radius of the circle?" src="https://miro.medium.com/v2/resize:fit:700/1*eLpJYFVM7OrDFg5UmwJeEQ.png" style="height:640px; width:700px" /> Figure 3: Rotated and flipped version of the problem Notice how √x has become x². That’s because rotating switches the x and y for each other. It problem suddenly looks much better! Now, let’s name some points of interest. These are shown in Figure 4. These points of interest are the following: the centre (a, b) and radius (r) of our circle (the smaller circle), and the three points of tangency ((a1, a2), (b1, b2) and (c1, c2)) of the three curves that form tangents with our circle respectively. <a href="https://dheerajdhobley.medium.com/another-purportedly-hard-geometry-puzzle-784c1b84b0a4">Click Here</a>