[(a^{2} - b^{2})tan(∠ROP)][a^{2}tan^{2}(∠ROP) +
b^{2}]' -
[a^{2}tan^{2}(∠ROP) + b^{2}][(a^{2} - b^{2})tan(∠ROP)]'
= 0, where the prime means differentiation with respect to ∠ROP.

Which is tan(∠ROP) = b/a and the slope(PQ) = -b/a.

Construction:

P is the intersection of the line (0,0)-(a,b) and Γ.

The tangent line is through P and parallel to the line (0,b)-(a,0).

QED

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