The locus of the foot of perpendicular drawn from the centre of the ellipse x² + 3y² = 6 on any tangent to it is:

Let the foot of perpendicular be P(h,k).
Equation of tangent with slope m passing P(h,k) is y = mx ± √(6m² + 2), where m = -h/k
⇒ √(6h²/k² + 2) = h² + k²/k
6h² + 2k² = (h² + k²)²
So required locus is 6x² + 2y² = (x² + y²)²