<s>[INST] Suppose $x^2+y^2=4$ and $z^2+t^2=9$ and $xt+yz=6$. What are $x,y,z,t$? Let $x^2+y^2=4$ and $z^2+t^2=9$ and $xt+yz=6$.
What are $x,y,z,t$?
 [/INST] Putting the parametric equation for both circles
$$x=2\sin a,y=2\cos a,z=3\sin b,t=3\cos b\\$$From this you get$$xt-yz=6\sin a\cos b+6\sin b\cos a=6\sin(a+b)=6\\\sin(a+b)=1\\a+b=\frac{\pi}{2}+2k
\pi$$
Which implies $x=2\sin(\pi/2-b)=2\cos b$ and $y=2\cos(\pi/2-b)=2\sin b$
From this we see that $(x,y,z,t)=(2\cos b,2\sin b,3\sin b,3\cos b)$ for each $b$
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