how can you tell by seeing $2ax+2by-2cz=0$ that every tangent plane passes through the origin? I am trying to solve this problem that asks to show every plane that is tangent to the cone $x^2+y^2=z^2$ passes through the origin. 
I saw other questions related because there were already some questions like this. But what I am wondering is something different. 
It goes as "set $f(x,y,z)=x^2+y^2-z^2$ then $$\nabla f(x,y,z)= <2x,2y,-2z>$$ Equation of tangent at point $P=(a,b,c)$ is $$\nabla f(a,b,c)=<2a,2b,-2c>$$ $$<2a,2b,-2c> \cdot <x-a,y-b, z-c> = 0$$ $$ 2ax-2a^2+2by-2b^2+(-2cz+2c^2)=0$$ $$ 2ax-2a^2+2by-2b^2-2cz+2c^2=0$$ $$2ax+2by-2cz-(2a^2+2b^2-2c^2)=0$$ $$2ax+2by-2cz=0$$
Now what I wonder is how the last equation $2ax+2by-2cz=0$ says "every tangent plane to the cone passes through the origin"? How can people by seeing this equation gather that every tangent plane passes through the origin?

Hint: Plug $x=y=z=0$ in your equation and you will see it.
