Question

if x y z are in h p then the value of (xz+xy-yz)(xy+yz-xz) /(xyz)²​

Answer 1

SOLUTION

GIVEN

x, y , z are in HP

TO DETERMINE

The value of

$\displaystyle \sf{ \frac{(xz + xy - yz)(xy + yz - xz)}{ {(xyz)}^{2} } }$

EVALUATION

Here it is given that x, y , z are in HP

$\displaystyle \sf{ \frac{2}{y} = \frac{1}{x} + \frac{1}{z} }$

$\displaystyle \sf{ \implies \: 2xz = yz + xy}$

Now

$\displaystyle \sf{ \frac{(xz + xy - yz)(xy + yz - xz)}{ {(xyz)}^{2} } }$

$\displaystyle \sf{ = \frac{(xz + xy + xy- 2xz)(2xz - xz)}{ {(xyz)}^{2} } }$

$\displaystyle \sf{ = \frac{(2 xy- xz).xz}{ {(xyz)}^{2} } }$

$\displaystyle \sf{ = \frac{(2y- z). {x}^{2}z }{ {x}^{2} {y}^{2} {z}^{2} } }$

$\displaystyle \sf{ = \frac{(2y- z) }{ {y}^{2} z } }$

$\displaystyle \sf{ = \frac{2}{yz} - \frac{1}{ {y}^{2} } }$

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