For a real variable a>1 , consider the points A_k= (k a, a^k ), k=1,2, ., n in the Cartesian plane. If and represent respectively the arithmetic mean of x -coordinates and the geometric mean of y coordinates of A_k , then the locus of the point P( , ) is

Step-by-step solution

We have, aligned & = a+2 a+3 a+ +n a n & = a[1+2+3+ +n] n & = a n(n+1) 2 n = a(n+1) 2 aligned aligned & and = (a a^2 a^3 a^4 a^n )^ 1 / n & = (a^ 1+2+3+ n )^ 1 / n =a^ ( n(n+1) 2 )^ 1 / n =a^ n+1 2 & Now, ^2=a^ n+1 & and ( 2 n+1 )^ n+1 = ( 2 a(n+1) 2(n+1) )^ n+1 =a^ n+1 & ^2= ( 2 n+1 )^ n+1 aligned So, locus of point P( , ) is y^2= ( 2 x n+1 )^ n+1