Let $S$ be the surface $x^2+2y^2+z^2=1$.
Find a parameterisation of $S$ and use it to find the equation of the tangent plane to $S$ at the point $\left(\frac1{\sqrt2},\frac12,0\right)$.
I can't work out an equation of tangent plane using parametrization onto polar coordinates. Is there any way to solve the question?
Hint: If
$$ x = x(\theta,\phi),\ y = y(\theta,\phi),\ z = z(\theta,\phi)\ $$
is your parametrization (which I assume is spherical coordinates, not polar), then the cross product of
$$ (\frac{\partial x}{\partial \theta}, \frac{\partial y}{\partial \theta}, \frac{\partial z}{\partial \theta}) $$
and
$$ (\frac{\partial x}{\partial \phi}, \frac{\partial y}{\partial \phi}, \frac{\partial z}{\partial \phi}) $$
is normal to the surface.