Let a tangent to the curve 9 x 2 + 16 y 2 = 144 intersect the coordinate axes at the points A and B . Then, the minimum length of the line segment A B is ______

Step-by-step solution

Given: 9 x 2 + 16 y 2 = 144 ⇒ x 2 16 + y 2 9 = 1 Equation of tangent at point P 4 cos θ ,   3 sin θ is x cos θ 4 + y sin θ 3 = 1 So, A is 4 sec θ , 0 and point B is 0 , 3 cosec θ . Hence, A B = 16   sec 2 θ + 9 cosec 2 θ ⇒ A B = 25 + 16 tan 2   θ + 9 cot 2   θ ⇒ 25 + 16 tan 2   θ + 9 cot 2   θ ≥ 7