The foot of the perpendicular drawn from either of the foci to any tangent to the ellipse S = 0 lies on a circle, concentric with the ellipse.

Let the equation of the ellipse be S = = 0.

• Let P(x₁, y₁) be the foot of the perpendicular drawn from either of the foci to a tangent.
• The equation of the tangent to the ellipse S = 0 is
  ----- (i)
• The equation to the perpendicular from either foci (±ae, 0) on this tangent is
  y = – (x ± ae) ----- (ii)
• As P is the point of intersection of (i) & (ii)
• We have y₁ = mx₁ ± , y₁ = – (x₁ ± ae)
• ⇒ y₁ – mx₁ = ± , my₁ + x₁ = ± ae
• ⇒ (y₁ – mx₁)² + (my₁ + x₁)² = a²m² + b² + a²e²
• ⇒ y₁² + m²x₁² – 2x₁y₁m + m²y₁² + x₁² + 2x₁y₁m = a²m² + a²(1 – e²) + a²e²
• ⇒ x₁²(m² + 1) + y₁²(1 + m²) = a²m² + a²
• ⇒ (x₁² + y₁²)(m² + 1) = a²(m² + 1)
• ⇒ x₁² + y₁² = a²
• ∴ P lies on x² + y² = a², which is a circle with center at the origin, the center of the ellipse.