Using componendo and dividendo, find the value of x, given \(\frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}}\; -\; 5} }}{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4{\rm{\;}}}\; - \;\sqrt {3{\rm{x}}\; - \;5} }}\) = 9.

Option 2 : 7

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10 Questions
10 Marks
7 Mins

\(\begin{array}{l} \frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}}\; - \;5} }}{{\sqrt {3{\rm{x\;}}+ {\rm{\;}}4{\rm{\;}}}\; - \;\sqrt {3{\rm{x}}\; - \;5} }} = 9\\ \Rightarrow {\rm{\;}}\frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} \;-\; 5} }}{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4{\rm{\;}}} \;-\; \sqrt {3{\rm{x}}\; - \;5} }}\; = \;\frac{9}{1} \end{array}\)

using componendo and dividendo, we get

\(\frac{{(\sqrt {3{\rm{x\;}} + {\rm{\;}}4} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} - 5)} {\rm{\;}} + {\rm{\;}}(\sqrt {3{\rm{x\;}} + {\rm{\;}}4} - \sqrt {3{\rm{x}} - 5)} }}{{(\sqrt {3{\rm{x\;}} + {\rm{\;}}4{\rm{\;}}} {\rm{\;}} + {\rm{\;}}\sqrt {3{\rm{x}} - 5)} - (\sqrt {3{\rm{x\;}} + {\rm{\;}}4} - \sqrt {3{\rm{x}} - 5)} }} = \frac{{9{\rm{\;}} + {\rm{\;}}1}}{{9 - 1}}{\rm{\;}}\)

\(\Rightarrow {\rm{\;}}\frac{{2\sqrt {3{\rm{x\;}} + {\rm{\;}}4} }}{{2\sqrt {3{\rm{x}} - 5} }} = \frac{{10}}{8} \Rightarrow \;\frac{{\sqrt {3{\rm{x\;}} + {\rm{\;}}4} }}{{\sqrt {3{\rm{x}} - 5} }} = \frac{5}{4}\)

\(\Rightarrow {\rm{\;}}\frac{{3{\rm{x\;}} + {\rm{\;}}4}}{{3{\rm{x}} - 5}} = \frac{{25}}{{16}}\) (On squaring both sides)

⇒ 75x – 125 = 48x + 64

⇒ 75x – 48x = 64 + 125

⇒ 27x = 189 ⇒ x = 7.