Question

If a-1/a =10 ; find : a. a+1/a , b. a²-1/a² ​

Answer 1

Answer:

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Step-by-step explanation:

$so \: here \: it \: is \: given \: that \\ a - \frac{1}{a} = 10 \\ squaring \: both \: sides \\ we \: get \\ a {}^{2} + \frac{1}{a {}^{2} } - 2.a \times \frac{1}{a} = 100 \\ ie \: \: a {}^{2} + \frac{1}{a {}^{2} } - 2 = 100 \\ so \: hence \: a {}^{2} + \frac{1}{a {}^{2} } = 102$

$so \: here \\ a {}^{2} + \frac{1}{a {}^{2} } \: is \: in \: form \: a {}^{2} + b {}^{2} \\ so \: we \: know \: that \\ a {}^{2} + b {}^{2} = (a + b) {}^{2} - 2ab \\ thus \: then \: accordingly \\ a {}^{2} + \frac{1}{a {}^{2} } = (a + \frac{1}{a} ) {}^{2} - 2.a. \frac{1}{a} \\ \\ 102 = (a + \frac{1}{a} ) {}^{2} - 2 \\ (a + \frac{1}{a}) {}^{2} = 104 \\ \\ taking \: square \: roots \: on \: both \: sides \: \\ we \: get \\ a + \frac{1}{a} = ±\sqrt{104} =± \: 2 \sqrt{26} \\ \\ so \: thus \: then \\ using \: \: a {}^{2} - \frac{1}{a {}^{2} } = (a + \frac{1}{a} )(a - \frac{1}{a} ) \\ = 2 \sqrt{26} \times 10 \\ = 20 \sqrt{26} \\ OR \\ = - 2 \sqrt{26} \times 10 \\ = - 20 \sqrt{26}$