1.

If a=5+2√6 , then find the value of a2+1/a2

Answer»

Identities USED :

{(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ \\ (a + b)(a - b) = {a}^{2} - {b}^{2}


Given,

a = 5+ 2√6

So,

\frac{1}{a} = \frac{1}{5 + 2 \sqrt{6} } \\

On rationalizing the DENOMINATOR we get,

\frac{1}{a} = \frac{1}{5 + 2 \sqrt{6} } \times \frac{5 - 2 \sqrt{6} }{5 - 2 \sqrt{6} } \\ \\ \frac{1}{a} = \frac{5 - 2 \sqrt{6} }{ {(5)}^{2} - {(2 \sqrt{6} )}^{2} } \\ \\ \frac{1}{a} = \frac{5 - 2 \sqrt{6} }{25 - 24} \\ \\ \frac{1}{a} = 5 - 2 \sqrt{6}

Now,

a + \frac{1}{a} = (5 + 2 \sqrt{6} ) + (5 - 2 \sqrt{6} ) \\ \\ a + \frac{1}{a} = 5 + 2 \sqrt{6} + 5 - 2 \sqrt{6} \\ \\ a + \frac{1}{a} = 10

Squaring both the SIDES we get,

{(a + \frac{1}{a} )}^{2} = {(10)}^{2} \\ \\ {a}^{2} + {( \frac{1}{a} )}^{2} + 2 \times a \times \frac{1}{a} = 100 \\ \\ {a}^{2} + \frac{1}{ {a}^{2} } = 100 - 2 \\ \\ { a }^{2} + \frac{1}{ {a}^{2} } = 98


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