Finding The Distance From A Paraboloid To A Plane.

[Baumer8993]

Homework Statement

Find the distance from the paraboloid z = X² + 2Y² to the plane
2X + 8Y + Z = -8.

Homework Equations

The partial derivatives with respect to X, And Y for the paraboloid.

The Attempt at a Solution

My professor said we need to find the point where the tangent plane of the paraboloid is parallel to the plane. I can take the X, and Y partial derivatives, but then I do not know what to do.

 

[Dick]

You should know what the normal vector to the plane is by looking at it. You want to find a point on the paraboloid whose normal vector is parallel to that. How would you find a normal vector to the surface at a point (x,y,z)?

 

[Baumer8993]

Oh I just need to move the Z over then take the gradient.

 

[Dick]

Oh I just need to move the Z over then take the gradient.

Yes, and the simple way to check if two vectors are parallel is to see if one is a multiple of the other.

 

[Baumer8993]

Ok, so now I am stuck at the finding the vectors that are parallel. I know that they can be a multiple of each other. I got the gradient of < 2X, 4y, -1>. I know the normal vector is
<2, 8, 1>.

I set the equations to make:

2X = 2K 4Y = 8K -1 = 1K

K is just a constant. How do I solve these?

 

[Dick]

Ok, so now I am stuck at the finding the vectors that are parallel. I know that they can be a multiple of each other. I got the gradient of < 2X, 4y, -1>. I know the normal vector is
<2, 8, 1>.

I set the equations to make:

2X = 2K 4Y = 8K -1 = 1K

K is just a constant. How do I solve these?

Good job setting the equations up, but I'm having a hard time figuring out why you can't solve -1=1*K. Take another look at them.