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Mathematics

If 5−65+6=a−b6\dfrac{5 - \sqrt{6}}{5 + \sqrt{6}} = a - b\sqrt{6}, find the values of 'a' and 'b'.

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Answer

Given,

Equation : 5−65+6=a−b6\dfrac{5 - \sqrt{6}}{5 + \sqrt{6}} = a - b\sqrt{6}

Rationalizing the denomiantor of L.H.S. of the above equation :

⇒5−65+6×5−65−6⇒(5−6)2(5)2−(6)2⇒(5)2+(6)2−2×5×625−6⇒25+6−10619⇒31−10619⇒3119−10619\Rightarrow \dfrac{5 - \sqrt{6}}{5 + \sqrt{6}} \times \dfrac{5 - \sqrt{6}}{5 - \sqrt{6}} \\[1em] \Rightarrow \dfrac{(5 - \sqrt{6})^2}{(5)^2 - (\sqrt{6})^2} \\[1em] \Rightarrow \dfrac{(5)^2 + (\sqrt{6})^2 - 2 \times 5 \times \sqrt{6}}{25 - 6} \\[1em] \Rightarrow \dfrac{25 + 6 - 10\sqrt{6}}{19} \\[1em] \Rightarrow \dfrac{31 - 10\sqrt{6}}{19} \\[1em] \Rightarrow \dfrac{31}{19} - \dfrac{10\sqrt{6}}{19}

Comparing 3119−10196 with a−b6\dfrac{31}{19} - \dfrac{10}{19}\sqrt{6} \text{ with } a - b\sqrt{6}, we get :

a=3119 and b=1019.a = \dfrac{31}{19} \text{ and } b = \dfrac{10}{19}.

Hence, a=3119 and b=1019a = \dfrac{31}{19} \text{ and } b = \dfrac{10}{19}.

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