Question

If a = 1/(a -5) where a not equal to 5 and a not equal to 0, find the values of: (a-1/a) (a+1/a) (a^2-1/a^2) (a^2+1/a^2)

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Answer 1

it is given that, a = 1/(a - 5) where a ≠ 5

we have to find the values of (a - 1/a), (a + 1/a), (a² - 1/a²) , (a² + 1/a²)

a = 1/(a - 5)

⇒a(a - 5) = 1

⇒a² - 5a = 1

⇒a²/a - 5a/a= 1/a

⇒a - 5 = 1/a

⇒a - 1/a = 5

using formula, (x + y)² = (x - y)² + 4xy

(a + 1/a)² = (a - 1/a)² + 4 × a × 1/a

= (5)² + 4

= 29

(a + 1/a) = √29

using formula, (x² - y²) = (x - y)(x + y)

(a² - 1/a²) = (a - 1/a)(a + 1/a)

= 5√29

therefore, (a² - 1/a²) = 5√29

using formula, x² + y² = (x + y)² - 2xy

so, (a² + 1/a²) = (a + 1/a)² - 2 × a × 1/a

= (a + 1/a)² - 2

= (√29)² - 2

= 29 - 2

= 27

therefore, (a² + 1/a²) = 27

Answer 2

Answer:

(i) 5

(ii) root over 29

(iii) 5 root over 29

(iv) 27