tangent plane of quadratic surfaceTangentPlaneOfQuadraticSurface2005-01-29 17:00:222006-11-01 12:48:25Resulttangent plane (elementary)% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe common equation of all quadratic surfaces in the rectangular $(x,\,y,\,z)$-coordinate system is
\begin{align}
Ax^2+By^2+Cz^2+2A'yz+2B'zx+2C'xy+2A''x+2B''y+2C''z+D = 0
\end{align}
where $A,\,B,\,C,\,A',\,B',\,C',\,A'',\,B'',\,C'',\,D$ are constants and at least one of the six first is distinct from zero.\, The equation of the tangent plane of the surface, with $(x_0,\,y_0,\,z_0)$ as the point of tangency, is
$$Ax_0x+By_0y+Cz_0z+A'(z_0y+y_0z)+B'(x_0z+z_0x)+C'(y_0x+x_0y)+
A''(x+x_0)+B''(y+y_0)+C''(z+z_0)+D = 0.$$
This is said to be obtained from (1) by polarizing it.
\textbf{Example.} \,The tangent plane of the {\em elliptic paraboloid}\, $4x^2+9y^2 = 2z$\, set in the point\, $(x_0,\,y_0,\,z_0)$\, of the surface is\, $4x_0x+9y_0y = z+z_0$,\, and especially in the point\, $(\frac{1}{2},\,\frac{1}{3},\,1)$\, it is\, $2x+3y -z-1 = 0$.