The tangent and normal to a rectangular hyperbola x y=c^2 at a point cuts off intercepts a₁ and a₂ on one axis and b₁, b₂ on the other, then a₁ a₂+b₁ b₂ is equal to

Step-by-step solution

Let the point be P (c t, c t ) The equation of the tangent to the hyperbola x y=c^2 at P is x c t +c t y=2 c^2 x+y t^2-2 c t=0...(i) Then, putting y=0, x=0 successively in Eq. (i), we get a₁=2 c t and b₁= 2 c t ( . intercepts of tangent on the axes are a₁ and b₁ ) Again the equation of the normal to the hyperbola x y=c^2 at P is x t^3-y t-c t^4+c=0 As before, a₂=c t- c t^3 and b₂=-c t^3+ c t a₁ a₂+b₁ b₂=2 c t (c t- c t^3 )+ 2 c t (-c t^3+ c t )=0