For 0 < θ < π 2 , four tangents are drawn at the four points ± 3 cos θ , ± 2 sin θ to the ellipse x 2 9 + y 2 4 = 1 . If A θ denotes the area of the quadrilateral formed by these four tangents, the minimum value of A θ is

Step-by-step solution

We have, Equation of ellipse x 2 9 + y 2 4 = 1 Equation of tangent at 3 cos θ , 2 sin θ is x 3 cos θ + y 2 sin θ = 1 Intercept of tangent is 3 sec θ , 0 and 0 , 2 cosec θ Area of quadrilateral = 4 × 1 2 3 sec θ × 2 cosec θ = 12 sec θ cosec θ = 24 sin 2 θ Minimum area = 24