Find value for a function

Question

I have computed $\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x+y}} = k$ (Where $k$ is some scalar).

I want to find the value of the following function in terms of $k$ and $m$:

$\frac{\sqrt{x} + \sqrt{my}}{\sqrt{x+my}}$ (Where $m$ is some scalar)

Please help if anyone can express the above function in terms of $k$ and $m$.

Answer 1

Divide the numerator and denominator of both expressions by $\sqrt{y}$. This will result in $$ k=\frac{\sqrt{x}+\sqrt{y}}{\sqrt{x+y}} =\frac{\sqrt{\frac{x}{y}} +1}{\sqrt{\frac{x}{y}+1}} $$ and $$ r=\frac{\sqrt{x}+\sqrt{my}}{\sqrt{x+my}} =\frac{\sqrt{\frac{x}{y}} +\sqrt{m}}{\sqrt{\frac{x}{y}+m}} $$ You can use the first equation to obtain the ratio $\frac{x}{y}$ in terms of $k$ and insert this ratio into the second equation.