# Two points in a strange shape

**Discrete Mathematics**Level 3

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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