If the tangent to the ellipse x 2 + 4 y 2 = 16 at P 4 cos ⁡ θ , 2 sin ⁡ θ is also a normal to the circle x 2 + y 2 = 8 x + 4 y , then the sum of all possible values of θ is equal to ∀ θ ∈ 0 , 2 π

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Correct answer: A. π 2

Tangent to the ellipse is x cos ⁡ θ + 2 y sin ⁡ θ = 4 which is a normal to the given circle ⇒ it satisfies 4 , 2 ⇒ sin ⁡ θ + cos ⁡ θ = 1 ∴ θ = 0 , π 2

If the tangent to the ellipse x 2 + 4 y 2 = 16 at P 4 cos ⁡ θ , 2 sin ⁡ θ is also a normal to the circle x 2 + y 2 = 8 x + 4 y , then the sum of all possible values of θ is equal to ∀ θ ∈ 0 , 2 π

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