Foot of perpendicular on a chord of a conic

Solution 1:

Okay to solve questions like these, let the chord be $y = mx + c$. Homogenize this chord with the ellipse to get a POSL. Since angle between lines is π/2, apply condition $coeff(x^2) + coeff(y^2) = 0$, to get $c = φ(m)$. Then put the condition that slope of FoP will be $-1/m$ and solve it with $y = mx + c$, and eliminate m to get locus.

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