Question

Please answer the following question ASAP
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Answer 1

Step-by-step explanation:

$given : a = \frac{1}{a - 5} \\ finding : \\ i) \: a - \frac{1}{a} = \frac{1}{a - 5} - \frac{1}{ \frac{1}{a - 5}} \\ = \frac{1}{a - 5} - (a - 5) \\ = \frac{1 - {(a - 5)}^{2} }{a - 5} \\ = \frac{1 - ( {a}^{2} - 10a + 25) }{a - 5} \\ = \frac{1 -{a}^{2} + 10a - 25) }{a - 5} \\ = \frac{ -{a}^{2} + 10a - 24}{a - 5} \\ = \frac{-({a}^{2} - 10a + 24)}{a - 5} \\ = \frac{-({a}^{2} - 6a - 4a + 24)}{a - 5} \\ = \frac{-(a(a - 6) - 4(a - 6))}{a - 5} \\ = \frac{ - (a - 4)(a - 6)}{a - 5} \\ ii) \: a + \frac{1}{a} = \frac{1}{a - 5} + \frac{1}{\frac{1}{a - 5}} \\ = \frac{1}{a - 5} + (a - 5) \\ = \frac{1 + {(a - 5)}^{2} }{a - 5} \\ = \frac{1 + {a}^{2} - 10a + 25}{a - 5} \\ = \frac{ {a}^{2} - 10a + 26}{a - 5} \\ iii) \: {a}^{2} - \frac{1}{ {a}^{2} } =(a - \frac{1}{a})(a + \frac{1}{a} ) \\ = - ( \frac{({a}^{2} - 10a + 24)}{a - 5})(\frac{ ( {a}^{2} - 10a + 26)}{a - 5}) \\ = - ( \frac{ {a}^{4 } - 10 {a}^{3} + 26 {a}^{2} - 10 {a}^{3} + 100 {a}^{2} - 260a + 24 {a}^{2} - 240a + 624}{ {(a - 5)}^{2} } \\ = - ( \frac{{a}^{4 }- 20 {a}^{3} + 150 {a}^{2} - 500a + 624}{ {(a - 5)}^{2} } )$

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