Question

Find the third proportional to x/y+y/x and x/y
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Answer 1

SOLUTION

TO DETERMINE

The third proportional to

$\displaystyle \sf{ \frac{x}{y} + \frac{y}{x} \: \: and \: \: \frac{x}{y} }$

EVALUATION

Let us take

$\displaystyle \sf{a = \frac{x}{y} + \frac{y}{x} = \frac{ {x}^{2} + {y}^{2} }{xy} \: \: \: and \: \: \: b = \frac{x}{y} }$

Let k be the third proportional

∴ a : b = b : k

$\displaystyle \sf{ \implies \: \frac{a}{b} = \frac{b}{k} }$

$\displaystyle \sf{ \implies \: ak = {b}^{2} }$

$\displaystyle \sf{ \implies \: k = \frac{ {b}^{2} }{a} }$

$\displaystyle \sf{ \implies \: k = \frac{ \frac{ {x}^{2} }{ {y}^{2} } }{ \frac{ {x}^{2} + {y}^{2} }{xy} } }$

$\displaystyle \sf{ \implies \: k = \frac{ {x}^{2} }{ {y}^{2} } \: . \: \frac{xy}{ {x}^{2} + {y}^{2} } }$

$\displaystyle \sf{ \implies \: k = \frac{ {x}^{3} }{y( {x}^{2} + {y}^{2}) } }$

FINAL ANSWER

Hence the required third proportional

$\displaystyle \sf{ = \frac{ {x}^{3} }{y( {x}^{2} + {y}^{2}) } }$

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