If y is the mean proportional between x and z, prove that: x2-y2+z2x-2-y-2+z-2=y4. - Mathematics

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`.

Sum

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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