Which one is equation of tangent

Question

Is equation of tangent plane

$z=f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $

or

$z=f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y_{0} ) $

In my book I found the first one, but on the internet I found second one too? Which of them is equation of tangent plane, what is the difference between these two?

Answer 1

You can verify the following condition yourself: The tangent at $(x_0, y_0, z_0)$ to a surface with equation $z = f(x, y)$ must touch the surface at the point $(x_0, y_0, z_0)$.

So because the point lies on the surface it satisfies the surface equation with $(x,y,z)$ replaced by $(x_0, y_0, z_0)$: $$ z_0 = f(x_0, y_0)$$

And it also needs to satisfy the tangent equation with $(x,y,z)$ replaced by $(x_0, y_0, z_0)$:

Case 1: $$z_0 = f(x_0, y_0) + f_x(x_0, y_0)\underbrace{(x_0-x_0)}_{=0} + f_y(x_0, y_0)\underbrace{(y_0-y_0)}_{=0} = f(x_0, x_0)$$

Case 2: $$z_0 = f_x(x_0, y_0)\underbrace{(x_0-x_0)}_{=0} + f_y(x_0, y_0)\underbrace{(y_0-y_0)}_{=0} = 0$$

So you can see case 2 only works in a special case where $f(x_0, y_0) = 0$

Answer 2

The second is the equation of a plane parallel to the tangent plane at the point $P=(x_0,y_0,f(x_0,y_0))$ to the surface $z=f(x,y)$ but passing thorough the origin.

The first is the equation of the tangent plane at the point$P$.

Substitute the coordinates $x_0$ , $y_0$ and $z_0=f(x_0,y_0)$ and you see that the first equation is verified, but the second is verified only if $z_0=f(x_0,y_0)=0$