Find the equation of the tangent plane of the surface

z^2 + x^2 − 4xy + y^2 = 2 at the point (1, 1, 2).

Question

Find the equation of the tangent plane of the surface

z^2 + x^2 − 4xy + y^2 = 2 at the point (1, 1, 2).

australopithecus · Answer

I know the equation is normally

$$z - z_0 = f_x(x_0,y_0)(x-x_0) - f_y(x_0,y_0)(y-x_0)$$

would I just use implicit differentiation to solve?

so take the derivative in respect to z?

anonymous · Answer

according to the equation u gave us we just need to computing  $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$

$z=f(x,y) $

for  $f_x(x_0,y_0)$  take partial derivative with respect to $x$ and for $f_y(x_0,y_0)$  take partial derivative wrt y

anonymous · Answer

and yes implicit differentiation...

australopithecus · Answer

so I got $$f_x(x,y) = \frac{-4x+2y}{-2z}$$
and
$$f_y(x,y) = \frac{2x-4y}{-2z}$$

australopithecus · Answer

I'm pretty sure i made a mistake

anonymous · Answer

yep

australopithecus · Answer

as I got 

z = (1/2)(x+y) +1

which seems incorrect

anonymous · Answer

see when u take derivative with respect to x u must treat y like a constant

anonymous · Answer

now u tell me what is $z'_y$

australopithecus · Answer

$$z_y = \frac{-4x + 2y}{-2z}$$

I got the same thing as last time

australopithecus · Answer

so (2x-y)/z

anonymous · Answer

yes thats right

anonymous · Answer

i made a little mistake too....lol

$$z'_x=\frac{2y-x}{z} $$

australopithecus · Answer

this is in the list of possible answers

A. 2z − x − y = 2 
B. 2z + x + y = 6 
C. z + x + y = 4 
D. z = 2 
E. 2x + 2y − z = 2 
F. z − x − y = 0

anonymous · Answer

o.O

i got 2z-x+y=4
what about u?

australopithecus · Answer

I got

x + y + 2= 2z

australopithecus · Answer

ugh my brain is so dead right now

australopithecus · Answer

2z - x - y = 2

anonymous · Answer

we're both wrong.....!!!

australopithecus · Answer

zx(x,y) = 1/2
zy(x,y) = 1/2

(1/2)(x-1) + (1/2)(y-1) + 2 = z

australopithecus · Answer

(1/2)(x-1) + (1/2)(y-1) + 2 = z
=
(1/2)x - 1/2 + (1/2)y - 1/2 + 2 = z
=
z - 1 =1/2(x+y)
=
2z - 2 = x+y
=
2z - x - y = 2

anonymous · Answer

oh .... there is a little typo in your first reply

z-z0=.....+.....(y-y0)  not  ........-.......(y-x0)

australopithecus · Answer

lol sorry about that I messed up the formula

australopithecus · Answer

none the less they should be the same x_0 = 1, y_0 = 1

australopithecus · Answer

so that formula works for this question

anonymous · Answer

so answer is 2z-x-xy=2

australopithecus · Answer

so it seems thanks for the help

anonymous · Answer

yw :)

australopithecus · Answer

Man I need sleep