**If a = 1/ (a – 5), find**

(i) a – 1/a

(ii) a + 1/a

(iii) a^{2} – 1/a^{2}.

**Answer :**

It is given that

a = 1/ (a – 5)

We can write it as

a^{2} – 5a – 1 = 0

Now divide each term by a

a – 5 – 1/a = 0

So we get

a – 1/a = 5

##### (i) a – 1/a = 5

##### (ii) We know that

**(a + 1/a) ^{2} = (a – 1/a)^{2} + 4**

Substituting the values

(a + 1/a)^{2} = 5^{2} + 4

So we get

(a + 1/a)^{2} = 25 + 4 = 29

a + 1/a = ± √29

##### (iii) We know that

**a ^{2} – 1/a^{2} = (a + 1/a) (a – 1/a)**

Substituting the values

a^{2} – 1/a^{2} = ± √29 × 5

a^{2} – 1/a^{2} = ± 5√29

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