I have a function for a plane: $x + z = d$ and a function for an ellipsoid $x^2 + y^2 + 2z^2 = 1$
The problem is to determine if there exists a number d, such that the plane is tangential to the ellipsoid.
I know that the normal for the plane is (1,0,1) and the gradient of the ellipsoid, which gives the normal, is (x, y, 2z). If the plane and the ellipsoid have the same normal vector at a point then they are tangential at that point.
Can I say that they both have the same normal vector at $x = 1, y = 0, z = \frac{1}{2}$ ?
There are some tips provided with the question. It first says that the function for the ellipsoid is "locally a function f of two variables". I'm not really sure about how I'm supposed to interpret this. Should I rearrange the function as $z = f(x,y)$?
It also says that I should determine the function for the tangent plane for the ellipsoid at the point $(a, b, f(a,b))$ and compare this with the plane x + z = d. This should give me an equation system containing a, b and d, but I haven't been able to get that equation system since I always end up with the dependent variable $f(a,b)$ in the equation which complicates things.