Planes perpendicular to vectors

[riskybeats]

a)Find the equation of the tangent plane to the graph z = x² + 4y² at the point (a, b, a² + 4b²)

b) For what values of a and b is the tangent plane perpendicular to the vector (3, -4, 2)?

I figured you have to use the tangent plane equation z - z_o = z_x(a,b)(x-a) + z_y(a,b)(y-b). This gave me the equation

Z = 2ax + 8by - a² - 4b²

For b, I think the dot product could be used if I set it to zero. But how would I go about solving for a and b? I think I may just be getting tired and a bit sloppy, because I know this is already stuff I have gone over, the only new thing is solving for a and b.

Any direction on this would be great, I'm just having trouble visualizing this problem.

 

[riskybeats]

I couldn't get to sleep because I kept on thinking this problem through. As far as I can see, if the plane's norm is the same as the vector <3, -4, 2>, then it must be perpendicular to it. I am not sure how to make the norm on the plane into that though based off of just of values of a and b. Does anyone have any suggestions?

 

[boboYO]

Doesn't have to be the same, just parallel, i.e. costheta=1 or -1.

Find the equation of the norm of the plane first.

 

[riskybeats]

The norm of my plane is influenced by a and b. This is what I have:

2a (X - a) + 8b (y - b) - z + a^2 + 4b^2 = 0

So if the vector is <3, -4, 2>, I could put the norm of my plane as <-3/2, 2, - 1> since that would be parallel? Hence a = -3/4 and b = 1/4 ?

 

[boboYO]

The norm of my plane is influenced by a and b. This is what I have:

2a (X - a) + 8b (y - b) - z + a^2 + 4b^2 = 0

So if the vector is <3, -4, 2>, I could put the norm of my plane as <-3/2, 2, - 1> since that would be parallel? Hence a = -3/4 and b = 1/4 ?

Hm, could you post the equation of the normal?

 

[riskybeats]

Hm, could you post the equation of the normal?

Sorry but I'm not sure what you mean by the equation of the normal. Do you mean the vector equation of the tangent plane?

<-3/2, 2, -1>$$\cdot$$<x + 3/4, y - 1/4, z> = 0?

Although I do still have those other numbers a² + 4b². Would that influence the normal of my plane still? I am tired and will probably think about this better later. Thanks for the help so far!