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प्रश्न
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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उत्तर
Since x : y is duplicate ratio of (x + z) and (y + z)
∴ x : y = (x + z)2 : (y + z)2
x(y + z)2 = y(x + z)2
On simplifying we get
xy2 + xz2 = yx2 + yz2
⇒ x2y - xy2 = xz2 - yz2
⇒ xy(x -y) = z2(x - y)
⇒ xy = z2
⇒ x : z : : z : y.
Hence proved.
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