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Question
If y is the mean proportional between x and z, prove that: `(x^2 - y^2 + z^2)/(x^(-2) - y^(-2) + z^(-2)) = y^4`.
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Solution
Given, y is the mean proportional between x and z.
`=>` y2 = xz
L.H.S = `(x^2 - y^2 + z^2)/(x^(-2) - y^(-2) + z^(-2))`
= `(x^2 - y^2 + z^2)/(1/x^2 - 1/y^2 + 1/z^2)`
= `(x^2 - xz + z^2)/(1/x^2 - 1/(xz) + 1/z^2)`
= `(x^2 - xz + z^2)/((z^2 - xz + x^2)/(x^2z^2))`
= x2z2
= (xz)2
= (y2)2 ...(∴ y2 = xz)
= y4
= R.H.S
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