The tangent to the ellipse x 2 25 + y 2 16 = 1 at point P lying in the first quadrant meets x -axis at Q and y -axis at R . If the length Q R is minimum, then the equation of this tangent is

Step-by-step solution

Let point P be 5 cos ⁡ θ , 4 sin ⁡ θ , θ ∈ 0 , π 2 So, the equation of tangent is x cos ⁡ θ 5 + y sin ⁡ θ 4 = 1 Points Q and R are 5 sec ⁡ θ , 0 and 0 , 4 cosec ⁡ θ respectively. Hence, Q R 2 = 25 sec 2 ⁡ θ + 16 cosec 2 ⁡ θ = 25 1 + tan 2 ⁡ θ + 16 1 + cot 2 ⁡ θ = 41 + 25 tan 2 ⁡ θ + 16 cot 2 ⁡ θ = 41 + 5 tan ⁡ θ - 4 cot ⁡ θ 2 + 40 Q R m i n 2 = 81 when tan 2 ⁡ θ = 4 5 ⇒ tan ⁡ θ = 2 5 Therefore, sin ⁡ θ = 2 3 , cos ⁡ θ = 5 3 Hence, equation of tangent is 2 x + 5 y = 6 5