Let two digit number is xy

The number is 10x + y

Given, two digits number is equal to 9 times the sum of its digits.

=> 10x + y = 9(x + y)

=> 10x + y = 9x + 9y

=> 10x + y - 9x - 9y = 0

=> x - 8y = 0 ..............1

Again, if 63 was subtracted from the number, the digits would be reversed.

=> 10x + y - 63 = 10y + x

=> 10x + y - 63 - 10y - x = 0

=> 10x + y - 10y - x = 63

=> 9x - 9y = 63

=> 9(x - y) = 63

=> x - y = 63/9

=> x - y = 7 ..............2

Subtract equation 1 and 2, we get

     -7y = -7

=> y = -7/-7

=> y = 1

From, equation 1, we get

      x - 8*1 = 0

=> x - 8 = 0

=> x = 8

So, the number is 81