Question
If $$\sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} {\text{,}}$$ where x and y are positive real numbers, then y is equal to ?
Answer: Option D
$$\eqalign{
& \Leftrightarrow \sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} \cr
& \Leftrightarrow x + \frac{x}{y} = {x^2}.\frac{x}{y} \cr
& \Leftrightarrow \frac{{xy + x}}{y} = \frac{{{x^3}}}{y} \cr
& \Leftrightarrow xy + x = {x^3} \cr
& \Leftrightarrow y + 1 = {x^2} \cr
& \Leftrightarrow y = {x^2} - 1 \cr} $$
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$$\eqalign{
& \Leftrightarrow \sqrt {x + \frac{x}{y}} = x\sqrt {\frac{x}{y}} \cr
& \Leftrightarrow x + \frac{x}{y} = {x^2}.\frac{x}{y} \cr
& \Leftrightarrow \frac{{xy + x}}{y} = \frac{{{x^3}}}{y} \cr
& \Leftrightarrow xy + x = {x^3} \cr
& \Leftrightarrow y + 1 = {x^2} \cr
& \Leftrightarrow y = {x^2} - 1 \cr} $$
Was this answer helpful ?
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