Level curves for the sum of the squares of the normals to an ellipse (Q434314)

Let \(c\) be an ellipse of the real Euclidean plane \({\mathbb R}^2\) with equation \(x^2/a^2+y^2/b^2=1\) and \(P=(x,y)\in{\mathbb R}^2\) any point. Assume that the point \(C\) varies on \(c\) and denote by \(n_C\) the normal at \(C\) to \(c\), then there exist (algebraically counted) four points \(C_1,\dots,C_4\) on \(c\) such that \(n_{C_1},\dots,n_{C_4}\) are incident with \(P\). For \(k=1,\dots,4\) put \(h_k\) for the (real or complex) distance from \(P\) to \(C_k\); the author calls \(h_k^2\) the ``square of the normal \(n_{C_k}\)''. (Reviewer's remark: For exactness one should speak of the ``four squared perpendicular distances between \(P\) and \(c\)''.) The author shows \[ \sum_{k=1}^{4}h_k^2=2\Bigl(a^4-b^4+(a^2-2b^2)x^2+(2a^2-b^2)y^2\Bigr)/(a^2-b^2). \] The locus of all points \((x,y)\in{\mathbb R}^2\) such that \(\sum_{k=1}^{4}h_k^2\) equals a given constant \(\gamma\in{\mathbb R}\) is called a \textit{level curve \(\ell_{\gamma}\) of \(c\)}. Depending on \(a\) and \(b\) the level curves to \(c\) are either ellipses or hyperbolas or pairs of straight lines. The author exhibits conditions for \(a\), \(b\), and \(\gamma\) such that 1) \(\ell_{\gamma}\) contains only inner points of \(c\), 2) \(\ell_{\gamma}\) is inscribed into the evolute of \(c\), and 3) \(\ell_{\gamma}\) touches \(c\). If \(c\) is a hyperbola, then a formula holds which is similar to the one above, but in the hyperbolic case each level curve is an ellipse.