From the question data, x and y are both positive numbers and x>y
The value of the expression needs to be evaluated.
From statement I alone, x + y = 4 + 2√xy.
Transposing the variable terms onto the LHS, we have x + y - 2√xy. = 4
The expression on the LHS represents the simplified form of\( (√x - √y)^2\). Therefore, the equation can be rewritten as \((√x - √y)^2\) = 4.
Taking the square root on both sides, we have (√x - √y) = ± 2; since the question says that both x and y are positive, (√x - √y) cannot be equal to – 2, therefore, (√x - √y) = 2.
In the expression given in the question stem, factoring out common terms, we have,
\(\frac{√2 (√x + √y )}{ (x-y)}\)
The denominator can be expressed as the product of (√x - √y) (√x + √y ).
Therefore, \(\frac{√2 (√x + √y )}{ (x-y)}\) = \(\frac{√2 (√x + √y )}{ (√x - √y) (√x + √y )}\)
Cancelling off (√x + √y ) and substituting the value of (√x - √y), value of the expression = \(\frac{√2 }{ 2}\) = \(\frac{1}{ √2}\)
Statement I alone is sufficient to answer the question. Answer options B, C and E can be eliminated.
From statement II alone, x – y = 9.
If x = 25 and y = 16, the value of the expression equal to √2
If x = 26 and y = 17, the value of the expression is not equal to √2
Statement II alone is insufficient to find a unique value for the given expression. Answer option D can be eliminated.
The correct answer option is A.
Hope that helps!
Aravind BT
_________________