Geometry + Kinematics Challenge

Consider the following gif (animation),

As seen from ground, the Orange ball is static whereas the Red Ball rotates about a fixed point and Also About itself (which has not been shown in animation) clock wise in both cases

Now suppose that it rotates about itself at \(w_1\) and rotates about the centre of the circle at \(w_2\) , Let the minimum distance between the two balls is \(\displaystyle D-R\) and the radius of the circle be R.

Then as seen by the moving ball or in the reference frame attached with the moving ball that also rotates along with it about itself,

The trajectory of the stationary ball can be written as the equation of a Conic, then the eccentricity can be written as

\(\frac {\sqrt a}{b}\)

Then find \( 35-(a+b)\)


1) \( \ D=3, { w }_{ 2 }=10, { w }_{ 1 }=5, R=2\)
All are in SI units.
2) \(a,b\) are all positive and real integers
3) The origin of the coordinate system in moving reference frame is the Red Ball itself


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