The locus of the foot of perpendicular drawn from the center

• Coordinate Geometry

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

1. (x² - y²)² = 6x² + 2y²
2. (x² - y²)² = 6x² - 2y²
3. (x² + y²)² = 6x² - 2y²
4. (x² + y²)² = 6x² + 2y²

Answer

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

The correct option is D.