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CAT 2017 [slot 2] Question with solution 28

Question 62:
How man Y different pairs(a,b) of positive integers are there such that $a\leq b$ and $\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$?
Answer: 3
Explanation:

$\frac{1}{a}+\frac{1}{b}=\frac{1}{9}$
=> $ab = 9(a + b)$
=> $ab - 9(a+b) = 0$
=> $ ab - 9(a+b) + 81 = 81$
=> $(a - 9)(b - 9) = 81, a > b$
Hence we have the following cases,
$ a - 9 = 81, b - 9 = 1$ => $(a,b) = (90,10)$
$ a - 9 = 27, b - 9 = 3$ => $(a,b) = (36,12)$
$ a - 9 = 9, b - 9 = 9$ => $(a,b) = (18,18)$
Hence there are three possible positive integral values of (a,b)


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