So I have a cone of the form $x^2+y^2 = k^2 z^2$ for some $k\in\mathbb{R}$ and some ellipse $\vec{x}(\theta)= \vec{c_0} + (\cos\theta)\, \vec{u} + (\sin\theta)\, \vec{v}$ parameterised by $\theta$, where $\vec{c_0}, \vec{u}, \vec{v}$ are fixed vectors in $\mathbb{R}^3$. I want to find out the conditions when the cone surface is tangent to the ellipse. How do I proceed?
(What I tried: Naively thought tangency in 3D space would imply tangent of curve A being along tangent of any curve on surface B)
Assume we have a surface given by $f(\vec x) = 0$ and a point $\vec x$ on that surface. Assuming $f$ is smooth then the points where the gradient of $f$ is non-zero are non-special. At those points the tangent plane to the surface consists of vectors along which the derivative of $f$ is zero. In our case $f(\vec x) = x^2 + y^2 - k^2 z^2$ and its gradient is non-zero everywhere except for $\vec x = \vec 0 = \begin{pmatrix}0\\0\\0\end{pmatrix}$. The gradient is given by $$\vec\nabla f = 2 \begin{pmatrix}x\\y\\-k^2z\end{pmatrix}.$$
In our case the curve is given in parametrized form by $\vec x(\theta)$, and the derivative along the curve is given by $$\frac{\partial \vec x(\theta)}{\partial \theta} = -\sin\theta \vec u + \cos \theta \vec v.$$ Thus, the system of equations for you to solve (except for tangent vectors at 0) is $$ \begin{cases} f(\vec x(\theta)) = 0,\\ \vec \nabla f(\vec x(\theta)) \cdot \frac{\partial \vec x(\theta)}{\partial \theta} = 0. \end{cases} $$ Note that for fixed surface and curve parameters ($k, \vec c_0, \vec u, \vec v$) this is a system of 2 equations for a single variable $\theta$, i.e. most ellipses are not tangent to the cone.
At the origin the definition of "tangent" could be ambiguous (depends on the exact wording on the definition of "tangent" you are using). For cone are 3 reasonable options: to say that none of the vectors are tangent, all vectors are tangent, and only those pointing "along" the cone are tangent. The first 2 are trivial, the last one boils down to the condition $w_x^2 + w_y^2 = k^2 w_z^2$ for a vector to be tangent (in our case we are interested in the vector $\vec w = \frac{\partial \vec x(\theta)}{\partial\theta}$ for $\theta$ s.t. $\vec x(\theta) = \vec 0$.