If y is the mean proportional between x and z ; show that xy + yz is the mean proportional between $x^{2} + y^{2}$ and $y^{2} + z^{2}$ .

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Solution

Since y is the mean proportion between x and z.
Therefore
$y^{2} = x z$

Now we have to prove that $x y + y z$ is the mean proportional between $x^{2} + y^{2}$ and $y^{2} + z^{2}$ i.e.

$(x y + y z)^{2} = (x^{2} + y^{2}) (y^{2} + z^{2})$

L. H. S
=> $(x y + y z)^{2}$
=> $y^{2} (x + z)^{2}$
=> $x z (x + z)^{2}$

R. H. S
=> $(x^{2} + y^{2}) (y^{2} + z^{2})$
=> $(x^{2} + x z) (x z + z^{2})$
=> $x (x + z) z (x + z)$
=> $x z (x + z)^{2}$
L. H. S = R. H. S
Hence proved

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