From any point on the hyperbola x^2 a^2 - y^2 b^2 =1 , tangents are drawn to the hyperbola x^2 a^2 - y^2 b^2 =2 . The area of the figure formed by the chord of contact of that point and the asymptotes is

Step-by-step solution

Let tangents are drawn from A(a, 0) to the hyperbola x^2 a^2 - y^2 b^2 =2P Q is the chord of contact. Asymptotes are A₁ and A₂ . Equation of A₁ and A₂ are x a - y b =0 and x a + y b =0 respectively. Equation of P Q is given by x x₁ a^2 - y y₁ b^2 =2 a x a^2 -0=2 x=2 a Now solving x=2 a and asymptotes A₁ and A₂ we get P (2 a, 2 b) and Q (2 a,-2 b) aligned & M is the mid-point of P Q & M (2 a, 0) & O M=2 a and P Q=4 b aligned Area ( O P Q)= 1 2 base height aligned & = 1 2 P Q O M= 1 2 4 b 2 a & =4 a b aligned