Two points in a strange shape

A strange shape can be made by joining two semi-ellipses with the same major axis but different minor axes. The above image is an example of one such shape. It is drawn in the complex plane as indicated by the axes.

The blue region is a semi-ellipse with semi-minor axis \((+Im)\) of length \(3\). The pink region is a semi-ellipse with semi-minor axis \((-Im)\) of length \(4\). The major axis \((Re)\), common to both semi-ellipses, has a length of \(10\).

Now, a complex number, \(z_{1}\), is randomly selected from the blue region, and another complex number \(z_2\) is randomly selected from the pink region.

Let \(P\) be the probability that \(\arg (z_1) = - \arg (z_ 2),\) then what is \(\lfloor 100P \rfloor?\)

Details and assumptions

  • \(|z_{1}| \neq 0\)

  • \(|z_{2}| \neq 0\)

  • \(\arg(z)\) is the argument of the complex number \(z.\) Its range is \((-\pi, \pi].\)

  • \(Re\) and \(Im\) represent the real and the imaginary axes, respectively.

This problem is a part of the set - A Strange Shape.

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