Let the tens and the units digits of the required number be x and y, respectively.
Then, we have:
xy = 14 ....(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴ (10x + y) + 45 = 10y + x
⇒ 9x − 9y = −45
⇒ 9(y − x) = 45
⇒ y − x = 5 ....(ii)
We know:
(y + x)2 − (y − x)2 = 4xy
⇒
∴ y + x = 9 ....(iii) (∵ x and y cannot be negative)
On adding (ii) and (iii), we get:
2y = 9 + 5 = 14
⇒ y = 7
On substituting y = 7 in (ii), we get:
7 − x = 5
⇒ x = (7 − 5) = 2
∴ Number = (10x + y) = 10 × 2 + 7 = 20 + 7 = 27
Hence, the required number is 27.