Math, asked by abhay110143, 9 months ago

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

Answers

Answered by pulakmath007
5

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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