I have to find the tangent line to the surface $$z=\frac{x^{2}}{4}+\frac{y^{2}}{9}$$ at $$P(2,-3,2)$$ such that this line pass through the $x$-axis.
I evaluated the directional derivative of $z$ in the direction of $\vec{u}=(2,-3)$ that is equal to $$\frac{4}{\sqrt{13}}$$ and so, the line equation is $$y+3=\frac{4}{\sqrt{13}}(x-2)$$ But I don't think this is the right way.
Hint. Find the plane tangent to the surface at the given point $P$. Such plane intersect the $x$-axis at $Q$. The line through $P$ and $Q$ is what you are looking for.