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 \( 35-(a+b)\)

Note:

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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