Let A θ 1 and B θ 2 be two points on the hyperbola x 2 a 2 - y 2 b 2 = 1 and S be the focus of the hyperbola. If A , S , B are collinear and a cos θ 1 + θ 2 2 = k cos θ 1 - θ 2 2 then k =

Step-by-step solution

Plotting the diagram of given data we get, We know that equation of chord joining A θ 1   &   B θ 2 is given by, x a cos θ 1 - θ 2 2 - y b sin θ 1 + θ 2 2 = cos θ 1 + θ 2 2 Now given this chord is passing through focus a e , 0 then, a e a cos θ 1 - θ 2 2 = cos θ 1 + θ 2 2 ⇒ e cos θ 1 - θ 2 2 = cos θ 1 + θ 2 2 ⇒ a a 2 + b 2 cos θ 1 - θ 2 2 = cos θ 1 + θ 2 2 ⇒ a cos θ 1 - θ 2 2