Consider surface x + x^2 + y^2 - 2z^2 = 1 and point (1,1,1)

[Xibalba]

I need some pointers / help w/ this problem:

Consider the surface x + x^2 + y^2 - 2z^2 = 1 and the point (1,1,1) which lies on the surface.

1) Find the equation of the tangent line to the surface at P.

2) Find an equation of the normal line to the surface at P.

For 1), I tried to take the derivative of the surface and got three partials:

fx = 1 + 2x
fy = 2y
fz = 4z

and then am I supposed to plug in P(1,1,1) into the equation a(x-x0) + b(y-y0) + c(z-z0) = 0 and use the partials for the x, y and z terms?

I have a feeling I've missed the concepts behind what the question is asking ...

 

[tkhunny]

Tangent "line"?! Perhaps you mean "plane".

1) Try evaluating the partial derivatives. These direction numbers should give you a clue.

2) Please observe that (x0,y0,z0) = (1,1,1)

 

[Xibalba]

Of course! These derivatives also serve as indicators of the direction of change. So, if you evaluate them at point P, they''re going to give you the directional vector(?)

Sorry, I meant to say the tangent plane to the surface, as the problem states.

Thanks for the observation. I shall now solve.

HUGE Thanks! Good night!