Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$

Question

Elliposid in $\mathbb{R}^3$ is given by $2x^2+2y^2+z^2=338$. Find radius of sphere that touches ellipsoid in points $(x,y,10)$. Sphere has center on $z-axis$

This is what I have so far.

Sphere should touch ellipsoid at circle $x^2+y^2=119$ where $z=10$. Sphere is given by formula $x^2+y^2+(z-z_0)^2=r$. We should determine $z_0$ and $r$. Sphere and ellipsoid should have same tangent planes in those points. Now I don't know how to find equations of tangent planes and place them in some equations. Can someone give me hints how to find tangent planes and check if they are same.

Answer 1

For example: the tangent plane to the ellipsoid $\:f(x,y,z)=2x^2+2y^2+z^2-338=0\;$ at some point $\;P:=(x_0,y_0,z_0)\;$ is given by

$$\frac\partial{\partial x}f(P)(x-x_0)+\frac\partial{\partial y}f(P)(y-y_0)+\frac\partial{\partial z}f(P)(z-z_0)=0$$

and in our case:

$$4x_0(x-x_0)+4y_0(y-y_0)+2z_0(z-z_0)=0\iff$$

$$ 4x_0x+4y_0y+2z_0z=2(2z_0^2+2y_0^2+z_0^2)=676(=2\cdot338)$$

From the above I think we could deduce the radius of sphere we're looking for is at $\;(0,0,10)\;$ and must have radius $\;\sqrt{119}\;$ , but you better check this carefully.