Q.
The arithmetic mean and the geometric mean of two distinct 2 digit numbers x and y are two integers one of which can be obtained by reversing the digits of the other (in base 10 representation ). Then, x+y equals
We have,
Let two-digits numbers are 10a+b
Given, 10a+b is AM of x and y
and 10b+a is GM of x and y ∴2x+y=10a+b ⇒xy=10b+a ⇒xy=(10b+a)2 ⇒(x+y)2−(x−y)2=4xy ∴(x−y)2=(x+y)2−4xy ⇒(x−y)2=4(10a+b+10b+a) (10a+b−10b−a) ⇒(x−y)2=4(11)(a+b)⋅9(a−b) ⇒(x−y)2=4×11⋅(a+b)⋅9(a−b) 4×11(a+b)×9(a−b) must be a perfect square ∴a+b=11,a−b=1
On solving these equations, we get a=6,b=5 ∴x+y=2(10a+b) ⇒x+y=2(60+5) ⇒x+y=130