The standard cartesian ellipse \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \) where \(a>b\) has the director circle \(x^2+y^2=a^2+b^2\).

The director circle, by definition, produces two tangents from any point on it that are perpendicular to each other. Now, at the points of contact, produce normals and let their point of intersection be \(L\). Let Z be the locus of all points L such that everything described as above is true. My question is simple, but messy. What is the algebraic relation that describes \(Z\), for any value of \(a\) and \(b\)?
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