tangent plane of quadratic surface

The common equation of all quadratic surfaces in the rectangular $(x,y,z)$-coordinate system is

$A{x}^2+B{y}^2+C{z}^2+2A^{\prime }yz+2B^{\prime }zx+2C^{\prime }xy+2A^{\prime \prime }x+2B^{\prime \prime }y+2C^{\prime \prime }z+D=0$ (1)

where $A,B,C,A^{\prime },B^{\prime },C^{\prime },A^{\prime \prime },B^{\prime \prime },C^{\prime \prime },D$ 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

$$A{x}_{0}x+B{y}_{0}y+C{z}_{0}z+A^{\prime }(z_{0}y+y_{0}z)+B^{\prime }(x_{0}z+z_{0}x)+C^{\prime }(y_{0}x+x_{0}y)+A^{\prime \prime }(x+x_0)+B^{\prime \prime }(y+y_0)+C^{\prime \prime }(z+z_0)+D=0.$$

This is said to be obtained from (1) by polarizing it.

Example.  The tangent plane of the elliptic paraboloid  $4x^2+9y^2=2z$  set in the point  $(x_0,y_0,z_0)$  of the surface is  $4x_{0}x+9y_{0}y=z+z_0$,  and especially in the point  $(\frac{1}{2},\frac{1}{3},\mathrm{\hspace{0.17em}1})$  it is  $2x+3y-z-1=0$.

Title	tangent plane of quadratic surface
Canonical name	TangentPlaneOfQuadraticSurface
Date of creation	2013-03-22 14:58:48
Last modified on	2013-03-22 14:58:48
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Result
Classification	msc 51N20
Related topic	TangentOfConicSection
Related topic	QuadraticSurfaces