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Hometangent plane of quadratic surface

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# tangent plane of quadratic surface

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

$\displaystyle Ax^{2}+By^{2}+Cz^{2}+2A^{{\prime}}yz+2B^{{\prime}}zx+2C^{{\prime% }}xy+2A^{{\prime\prime}}x+2B^{{\prime\prime}}y+2C^{{\prime\prime}}z+D=0$ | (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

$Ax_{0}x+By_{0}y+Cz_{0}z+A^{{\prime}}(z_{0}y+y_{0}z)+B^{{\prime}}(x_{0}z+z_{0}x% )+C^{{\prime}}(y_{0}x+x_{0}y)+A^{{\prime\prime}}(x+x_{0})+B^{{\prime\prime}}(y% +y_{0})+C^{{\prime\prime}}(z+z_{0})+D=0.$ |

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 $(\frac{1}{2},\,\frac{1}{3},\,1)$ it is $2x+3y-z-1=0$.

## Mathematics Subject Classification

51N20*no label found*

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