Question

If the duplicate ratio of a+x:b+x is a:b then show that x=+√ab ​

Answer 1

Answer:

(Check the image)

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❤ HOPEFULLY IT HELPS ❤

Answer 2

❖ Given :

• If the duplicate ratio of a + x : b + x is a : b then show that x = ±√ab

❖ Solution :

Since, duplicate ratio of a + x : b + x is a : b .

$\\ \therefore \tt \: \: \: \: \dfrac{(a + x {)}^{2} }{ {(b + x)}^{2} } = \frac{a}{b} \\ \\ \\ \tt : \implies \: \frac{ {a}^{2} + 2ax + {x}^{2} }{ {b}^{2} + 2bx + {x}^{2} } = \frac{a}{b}$

By doing cross multiplication we get :

$\\ \tt : \implies \: b( {a}^{2} + 2ax + {x}^{2} ) = a( {b}^{2} + 2bx + {x}^{2} ) \\ \\ \\ \tt \: : \implies \: {a}^{2} b +2abx + b {x}^{2} = a{b}^{2} + 2abx + a{x}^{2}$

Cancelling 2abx both sides :

$\tt : \implies \: {a}^{2} b + \cancel{2abx} + b {x}^{2} = a{b}^{2} + \cancel{2abx }+ a{x}^{2} \: \\ \\ \\ \tt : \implies {a}^{2} b + b {x}^{2} = a {b}^{2} + a {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \tt \: : \implies {a}^{2} b - a {b}^{2} = a {x}^{2} - b {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \tt \: : \implies ab(a - b) = {x}^{2} (a - b) \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Cancelling (a - b) from both sides :

$\\ \tt \: : \implies ab \cancel{(a - b) }= {x}^{2} \cancel{ (a - b)} \\ \\ \\ \tt : \implies \: ab = {x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \tt : \implies \: { \underline{ \boxed{ \frak{ \pmb{x = ± \sqrt{\: ab}}}} }} \: \: \purple{\bigstar} \: \: \: \: \: \: \: \: \:$