A normal is drawn to the ellipse x 2 9 + y 2 = 1 at the point 3 cos ⁡ θ , sin ⁡ θ where 0 < θ < π 2 . If N is the foot of the perpendicular from the origin O to the normal such that O N = 2 , then θ is equal to

Step-by-step solution

Equation of the normal at 3 cos θ , sin θ is 3 x cos ⁡ θ - y sin ⁡ θ = 8 So, O N = - 8 9 sec 2 ⁡ θ + cosec 2 ⁡ θ = 2 (given) ⇒ 16 = 9 1 + tan 2 ⁡ θ + 1 + cot 2 ⁡ θ ⇒ 6 = 6 + 3 tan ⁡ θ - cot ⁡ θ 2 ⇒ 3 tan ⁡ θ - cot ⁡ θ = 0 Hence, tan 2 ⁡ θ = 1 3 ⇒ θ = π 6