Question

$if \\ \: 5( \frac{1}{ {x}^{2} } + \frac{1}{ {y}^{2} } + \frac{1}{ {z}^{2} } ) = 4( \frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} ) \\ find \: \\ \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$
Answer It.
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Answer 1

||✪✪ QUESTION ✪✪||

if 5(1/x² + 1/y² + 1/z²) = 4(1/xy + 1/yz + 1/zx)

Find (1/x + 1/y + 1/z) ?

|| ✰✰ ANSWER ✰✰ ||

Let :-

➼ 1/x = a

➼ 1/y = b

➼ 1/z = c

Putting we get,

➺ 5a² + 5b² + 5c² = 4(ab+bc+ca)

Now, Splitting Them and taking RHS , in LHS side we get,

➺ a² + 4a² + b² + 4b² + c² + 5c² - 4ab - 4bc - 4ac = 0

Re-arranging LHS part now,

➺ (a² + 4b² - 4ab) + (4a²+c² - 4ac) + (b² + 4c² - 4bc)

= 0

➺(a-2b)² + (2a-c)² + (b-2c)² = 0

Putting First Two Equal to zero now, we get,

☛ a= 2b = c/2

But

☛ b= 2c

Hence, we can say That,

☛ a = 0 = b = c

So, (a+b+c) = ( 1/x + 1/y + 1/z) = 0

✪✪ Hence Proved ✪✪

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☙❦ For One More Solution Refer To Image Now. ❦☙

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Answer 2

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$

Hence proved.