## Calculus 8th Edition

Every tangent plane to the cone passes through the origin $(0,0,0)$.
Our aim is to determine the tangent plane equation for an ellipsoid. The general form is: $(x_2-x_1)f_x(x_1,y_1,z_1)+(y_2-y_1)f_y(x_1,y_1,z_1)+(z_2-z_1)f_x(x_1,y_1,z_1)=0$ ...(1) From the given data, we have $f(x,y,z)=(x_0,y_0,z_0)$ Equation (1), becomes: Thus, $(x-x_0)(2x_0)+(y-y_0)(2y_0)-(z-z_0)(2z_0)=0$ After simplifications, we get $xx_0+yy_0-zz_0=x_0^2+y_0^2-z_0^2$ Tangent plane is passing through the origin $(0,0,0)$, therefore $xx_0+yy_0-zz_0=0$ It is proved that every tangent plane to the cone passes through the origin $(0,0,0)$.