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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?

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  • Polar coordinates? If my guess on what you tried to do is right, there's a small scaling to make that work. Can you show us your parametrization? – N.Bach May 12 '17 at 14:05

1 Answers1

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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.