If tangent at a point P₁ (other than (0,0) ) on the curve y^2=a x^3 meets the curve again at P₂ . The tangent at P₂ meets the curve again at P₃ and so on, then find _ n _ i=1 ^n x_i , where x_i^ s are abscissae of P_i with x₁=3 .

Step-by-step solution

Let P₁= (t₁^2, a t₁^3 ) and P₂= (t₂^2, a t₂^3 ) aligned & 2 y d y d x =3 a x^2 d y d x = 3 a x^2 2 y & Slope of tangent at P₁= 3 a t₁ 2 = a (t₁^3-t₂^3 ) (t₁^2-t₂^2 ) & 3 (t₁ ) (t₁+t₂ )=2 (t₁^2+t₂^2+2 t₁ t₂ ) & t₁^2-t₂^2+t₁ t₂=0 (t₁+2 t₂ ) (t₁-t₂ )=0 aligned t₁=-2 t₂ Abscissae, t₁^2, t₂^2, t₃^2, . t₁^2, t₁^2 4 , t₁^2 16 , . will be G.P. with common ratio 1 4 for a . Sum of abscissae = x₁ 1- 1 4 = 3 3 / 4 =4