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# 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 - ICSE Class 10 - Mathematics

#### 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

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#### APPEARS IN

Selina Solution for Selina ICSE Concise Mathematics for Class 10 (2018-2019) (2017 to Current)
Chapter 7: Ratio and Proportion (Including Properties and Uses)
Ex.7B | Q: 12

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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.
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