MHB Axis Constrained Collision Distance

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To determine the distance until contact between two circles, draw a line from the center of the larger circle in the direction of its movement. Calculate the perpendicular distance from the center of the smaller circle to this line. This distance is crucial for finding the exact point of contact. Without this information, solving the problem is not feasible. Accurate calculations are essential for precise positioning of the circles.
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I have 2 curves and I need to slide 1 of the circles along an axis until it exactly meets the other circle. How do I determine the distance until contact?
 

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Draw a line from the center of the larger circle in the direction of its travel. You need to know the perpendicular-line distance from the center of the smaller circle to the drawn line. It is impossible to solve the problem without some information along those lines.
 

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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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