p, x₁, x₂ , x_n and q, y₁, y₂, , y_ n are two arithmetic progressions with common differences a and b respectively. If and are the arithmetic means of x₁, x₂, x_n , and y₁, y₂, , y_n respectively. Then the locus of P( , ) is

Step-by-step solution

It is given that p, x₁, x₂, x₃ x_n and q, y₁, y₂, y₃ y_n are in A.P. whose common difference are a and b respectively. aligned x₁ & =p+a, x_n=p+n a y₁ & =q+b, y_n=q+n b aligned Also, given is A.M. of x₁, x₂, x₃ x_n array rlrl & & = x₁+x₂+x₃+ .+x_n n & = n 2 (x₁+x_n ) n & = x₁+x_n 2 array Similarly, is A.M. of y₁, y₂, y₃ y_n . So, = y₁+y_n 2 Thus, substituting the value of x, x_n, y and y_n , we get From Eqs. (i) and (ii) eliminate (n+1) , we get aligned & 2 -2 p a = 2 -2 q b & b( -p)=a( -q) aligned Hence, locus of