Discussion

Explanation:

Given $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$

$\frac{b+a}{ab}=\frac{1}{9}$

⇒ 9b + 9a = ab
⇒ ab - 9b – 9a = 0
⇒ ab - 9b – 9a + 81 = 81 [Adding 81 both sides]
⇒ b(a – 9) – 9(a – 9) = 81
⇒ (a – 9) × (b – 9) = 81

Now 81 as a product of 2 positive integers can be written as 1 × 81, 3 × 21 or 9 × 9

If a - 9 = 1 ⇒ a = 10.
Also b - 9 = 81 ⇒ b = 90

Further, if a – 9 = 3, ⇒ a = 12
Also b – 9 = 27 ⇒ b = 36

If a – 9 = 9 ⇒ a = 18.
​​​​​​​Also b – 9 = 9 ⇒ b = 18

∴ 3 pairs of number i.e., (10, 90), (12, 36) and (18, 18) will satisfy the given equation.

Hence, 3.

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