|
The common equation of all quadratic surfaces in the rectangular $(x,\,y,\,z)$ -coordinate system is
 |
(1) |
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.
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 $4x_0x+9y_0y = z+z_0$ , and especially in the point $(\frac{1}{2},\,\frac{1}{3},\,1)$ it is $2x+3y -z-1 = 0$ .
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)