Question : If $x+ \sqrt{5} = 5+\sqrt{y}$ are positive integers, then the value of $\frac{\sqrt{x}+y}{x+ \sqrt{y}}$ is:
Option 1: 1
Option 2: 2
Option 3: $\sqrt{5}$
Option 4: 5
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Correct Answer: 1
Solution :
Given:
$x+ \sqrt{5} = 5+\sqrt{y}$
By comparing rational and irrational numbers, we get
⇒ $x = 5$ and $\sqrt{y} = \sqrt{5}$ ⇒ $y = 5$
By putting the values of $x$ and $y$ in the given expression,
$ \frac{\sqrt{x}+y}{x+ \sqrt{y}}$
$=\frac{\sqrt{5}+5}{5+ \sqrt{5}}$
$= 1$
Hence, the correct answer is 1.
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