If tangent are drawn to the ellipse x² + 2y² = 2 at all points on the ellipse other than its four vertices, then the midpoints of the tangents intercepted between the coordinate axes lie on the curve:

The correct option is C
1/2x² + 1/4y² = 1
Equation of general tangent on ellipse
xa secθ + yb cosecθ = 1
a = √2, b = 1 ⇒ x√2 secθ + y cosecθ = 1
Let the midpoint be (h,k)
h = √2 secθ / 2 ⇒ cosθ = 1/(√2h)
and k = cosecθ / 2 ⇒ sinθ = 1/(2k).
∴ sin²θ + cos²θ = 1 ⇒ 1/(2h)² + 1/(4k)² = 1
∴ 1/2x² + 1/4y² = 1