Consider the hyperbola H : x 2 - y 2 = 1 and a circle S with center N x 2 , 0 . Suppose that H and S touch each other at point P x 1 , y 1 with x 1 > 1 a n d y 1 > 0 . The common tangent to H and S at P intersects the x - axis at point M. If ( l , m ) is the centroid of the triangle Δ P M N , then the correct expression(s) is(are)

Step-by-step solution

Equation of tangent at P on hyperbola. x x 1 - y y 1 = 1 Point M 1 x 1 , 0 Equation of normal at P x x 1 + y y 1 = 2 Since ( x 2 , 0 ) satisfies its x 2 = 2 x 1 Centroid l , m ≡ x 1 + 1 3 x 1 , y 1 3 d l d x 1 = 1 - 1 3 x 1 2 d m d y 1 = 1 3 x 1 2 - y 1 2 = 1 y 1 = x 1 2 - 1 m = x 1 2 - 1 3 d m d x 1 = x 1 3 x 1 2 - 1 Alternative Method l = 3 sec ⁡ θ + cos ⁡ θ 3 m = tan ⁡ θ 3 = y 1 3 d l d x 1 = 3 sec ⁡ θ tan ⁡ θ - sin ⁡ θ 3 sec ⁡ θ tan ⁡ θ = 1 - 1 3 s e c 2 θ = 1 - 1 3 x 1 2 d m d x 1 = sec 2 ⁡ θ 3 sec ⁡ θ tan ⁡ θ