If $(x,y)$ satisfies the simultaneous equations \begin{align} 3xy - 4x^2 - 36y + 48x &= 0, \\ x^2 - 2y^2 &= 16, \end{align}where $x$ and $y$ may be complex numbers, determine all possible values of $y^2$.

[tex]3xy-4x^2-36y+48x=3y(x-12)-4x(x-12)=(3y-4x)(x-12)=0[/tex]

So either [tex]3y=4x[/tex], or [tex]x=12[/tex]. In the first case, we find

[tex]x^2-2y^2=16\implies x^2-2\left(\dfrac{4x}3\right)^2=16\implies x^2=-\dfrac{144}{23}[/tex]

from which it follows that

[tex]-\dfrac{144}{23}-2y^2=16\implies y^2=-\dfrac{256}{23}[/tex]

Alternatively, if [tex]x=12[/tex], then

[tex]12^2-2y^2=16\implies y^2=64[/tex]

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If $(x,y)$ satisfies the simultaneous equations \begin{align*} 3xy - 4x^2 - 36y + 48x &= 0, \\ x^2 - 2y^2 &= 16, \end{align*}where $x$ and $y$ may be complex numbers, determine all possible values of $y^2$.

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If $(x,y)$ satisfies the simultaneous equations \begin{align} 3xy - 4x^2 - 36y + 48x &= 0, \\ x^2 - 2y^2 &= 16, \end{align}where $x$ and $y$ may be complex numbers, determine all possible values of $y^2$.