I have two problems that don't seem to match. The task is to find the parametric equation for the normal line to the tangent plane. I'm seeing that the vector value of "z" is (-1) with one formula, but not in the other. These two seem to be the same, but the first was solved implicitly in the book (?) and arrived at (-1). The second was solved easily with the second formula, but arrived at a unique value for z.
The normal line to (x^2)+(y^2)+(z^2)=25 at p(-3,0,4)
is x=-3+(3t/4) y=0 z=4-t
The normal line to (x^2)+(y^2)+4(z^2)=12 at p(2,2,1)
is x=2+t y=2+t z= 1+2t
it makes sense that z= f(x.,y.)-t
different values for z seem ok, such as z= z.+fz(x.,y.,z.)
The Attempt at a Solution
The two problems seem like they are nearly the same and should both be solved with the second formula. I'm completly lost trying to figure out what I'm missing.