If the line y = x tan θ cuts the curve x 3 + x y 2 + 2 x 2 + 2 y 2 + 3 x + 1 = 0 at the points A ,   B and C . If O A ,   O B and O C are in HP , then tan θ is equal to

Step-by-step solution

Given, line y = x tan θ cuts the curve x 3 + x y 2 + 2 x 2 + 2 y 2 + 3 x + 1 = 0       … i at three points A , B and C . Let us assume A x 1 , x 1 tan θ , B x 2 , x 2 tan θ and C x 3 , x 3 tan θ ⇒ O A = x 1 2 + x 1 2   tan 2 θ = x 1 1 + tan 2 θ = x 1 sec θ Similarly O B = x 2 sec θ and O C = x 3 sec θ ∵ O A , O B , O C in H . P . ⇒ O B = 2 · O A O C O A + O C ⇒ x 2 · sec θ = 2 x 1 · x 3 sec 2 θ x