#### 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`

#### Solution

Given, y is the mean proportional between x and z.

`=> y^2 = xz`

LHS =` (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)` (∵ `y^2 = xy`)

`= (x^2 - xz = z^2)/((z^2 - xz + x^2)/(x^2z^2))`

`= x^2z^2`

`= (xz)^2`

`= (y^2)^2` ` (∴ y^2 = xz)`

`= y^4`

= R.H.S

Is there an error in this question or solution?

Solution 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 Concept: Proportions.