= Solution
\b[The printed point is not on the printed surface, so the requested tangent plane does not exist.] At $(2,-2,9)$, the left side of the defining equation is $x^2y=-8$, whereas the right side is $4ze^{x+y}-11=25$. This is a direct counterexample to the premise that the stated point belongs to the <surface>.
To make the obstruction precise, set $G(x,y,z)=x^2y-4ze^{x+y}+11$. The specified <surface> is the <level set> $G=0$, but $G(2,-2,9)=-33$. Although $\nabla G(2,-2,9)=(-44,-32,-4)$, the resulting plane $11x+8y+z=15$ is tangent to the different <level set> $G=-33$ through that point. It is not an answer for the printed <surface>. A <tangent plane requires the point to lie on the surface>.
If the intended correction preserves $x=2,y=-2$, the surface equation instead forces $z=3/4$. At the corrected point $p_*=(2,-2,3/4)$,
$$
\nabla G(p_*)=(-11,1,-4).
$$
The corresponding conditional <tangent plane> is
$$
\boxed{11x-y+4z=27\qquad\text{at the corrected point }(2,-2,3/4).}
$$
This conditional calculation identifies a possible repair; it does not silently replace the source's coordinate $9$.
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