Tangent plane equation question.

[Pavoo]

Homework Statement

Consider a surface ω with equation:

$$x^2 + y^2 + 4z^2 = 16$$

Find an equation for the tangent plane to ω at point (a,b,c).

Homework Equations

Tangent plane, 3 variables:

$$f_{1}(a,b,c)(x-a) + f_{2}(a,b,c)(y-b) + f_{3}(a,b,c)(z-c)= 0$$

The Attempt at a Solution

I get at the end:

$$ax + by + 4cz = a^2 + b^2 + 4c^2$$

The textbook gives me:

$$ax + by + 4cz = a^2 + b^2 + 4c^2 = 16$$

Where does the 16 come from?

Comparing to this problem, as an example:

Find an equation of the tangent plane to the sphere $$x^2 + y^2 + z^2 = 6$$ at point (1,-1,2). This on is simple. But not the first above.

And is it possible to solve this by expanding to a fourth variable, such as ω?

 

[voko]

The (a, b, c) point must satisfy the equation of the surface, hence the right-hand side equals 16.