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Is it possible to figure out 2 unknowns from only their ratio and the sum of their squares??

Hi, apologies if this is a stupid question - I haven't done maths proper since I left school a few years ago. 

Is it possible to figure out two unknowns (x and y) if I know their ratio to each other and the sum of their squares? 

What I have is as follows:

16:9 = x:y

1806336 = (x\^2 + y\^2)

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I'm trying to figure out things related to my laptop's screen resolution. It is 14" diagonally and the resolution is stated as 1600x900, for which the pixels per inch should be around 131. But the command "wmic desktopmonitor get PixelsPerYLogicalInch" returns a value of 96. 

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Needless to say I am confused and appreciate anyones help.  Thank you.
gpt
Sure. Divide both sides by the square of one of the variables.

1806336/y^(2) = (x^(2) + y^(2))/y^(2) = (x/y)^(2)+1

Since we know the ratio x/y we find

1806336/y^(2) = (16/9)^(2)+1=337/81

We then find y^(2)=1806336(81/337)

From this you solve for x^(2)=1806336(1 - 81/337)