Question

11. If x and y be unequal and x : y is the
duplicate ratio of x + 3 and y + z prove that
z is mean proportional between x and y. plz help fast​

Answer 1

x : y is the duplicate ratio of x + z and y+ z

Therefore  x:y = (x+z)2:(y+z)2 

              x/y = (x+z)2/(y+z)2 

             x(y+z)2 = y(x+z)2 

   x(y2 + z2 + 2yz) = y(x2 + z2 + 2xz)

xy2 + xz2 + 2xyz = yx2 + yz2 + 2xyz

xy2 + xz2 = yx2 + yz2 

xy2 - yx2 = yz2 - xz2 

xy(y - x) = z2(y-x)

y ≠ x because given x and y are unequal in the problem.

Therefore xy = z2

Therefore z is mean proportional between x and y.