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Question
If x and y be unequal and x : y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.
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Solution
Given, `x/y = (x + z)^2/(y + z)^2`
`=> x/y = (x^2 + 2xz + z^2)/(y^2 + 2yz + z^2)`
`=> x(y^2 + 2yz + z^2) = y (x^2 + 2xz + z^2)`
`=> xy^2 + 2xyz + xz^2 = x^2y + 2xyz + yz^2`
`=> xy^2 - x^2y = yz^2 - xz^2`
`=> xy(-x + y) = z^2 (y - x)`
`=> z^2 = xy`
`=> x/z = z/y`
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