Question

यदि a:b=c:d हो तो k नियम दारा सिंधद कीजिए।
a2+c2. ac
-------------- = -------
b2+d2. bd​

Answer 1

समाधान :

दिया हुआ :

a : b = c : d

सिद्ध करने के लिए :

$\displaystyle \sf{ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } = \frac{ac}{cd} \: }$

प्रमाण :

यहाँ यह दिया गया है कि

$\sf{a : b = c : d}$

$\displaystyle \sf{ \frac{a}{b} = \frac{c}{d} = k \:(say) }$

$\therefore \: \sf{ a = bk}$ और $\sf{c = dk \: }$

LHS

$\displaystyle \sf{ \frac{ {a}^{2} + {c}^{2} }{ {b}^{2} + {d}^{2} } \: }$

$= \displaystyle \sf{ \frac{ {(bk)}^{2} + {(dk)}^{2} }{ {b}^{2} + {d}^{2} } \: }$

$= \displaystyle \sf{ \frac{ {b}^{2} {k}^{2} + {d}^{2} {k}^{2} }{ {b}^{2} + {d}^{2} } \: }$

$= \displaystyle \sf{ \frac{ {k}^{2} ( {b}^{2} + {d}^{2}) }{ {b}^{2} + {d}^{2} } \: }$

$= \displaystyle \sf{ {k}^{2} \: }$

RHS

$= \displaystyle \sf{ \frac{ac}{bd} \: }$

$= \displaystyle \sf{ \frac{bk.dk}{bd} \: }$

$= \displaystyle \sf{ \frac{bd{k}^{2} }{bd} \: }$

$= \displaystyle \sf{ {k}^{2} }$

LHS = RHS

Hence proved

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