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Orthogonal to the line from the centroid to the circumscribed cylinder

I’ve been wondering about how to approach the following problem:
Given the centroid $C=(x_C,y_C,z_C)$ and the inscribed cylinder of the paraboloid $z=(x-x_C)^2 +(y-y_C)^2$.
Find a plane that is orthogonal to the line from the centroid to the circumscribed cylinder.
As much as I understand it, the plane has to be on the line $g_1:Cx+Cy+Dz=0$. But how can I get to it?

A:

Solve the system.
Multiply the first equation by $x$ and then by $y$, to get the system
$$\begin{cases} x_Cx-xx_C+Cx&=0\\ y_Cx+xy_C-x^2-Cy^2&=0\,, \end{cases}$$
or, collecting terms,
$$x_C=0\,,\qquad y_C=0\,;\quad C=(2x,2y)\,.$$
Now solve for $D$.

Hisathala Vinayagam

Hisathala Vinayagam is a 1979 Indian Malay

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