# Point on paraboloid at which the tangent plane is parallel to plane

• Oct 3rd 2011, 07:17 AM
deezy
Point on paraboloid at which the tangent plane is parallel to plane
Find pt on paraboloid $x = 5y^2 + 7z^2$, if it exists, at which the tangent plane is parallel to plane $-x + y + z = 3$.

Not completely sure how to approach this problem. I'm pretty sure it involves the gradient, so I set $f(x,y,z) = x - 5y^2 - 7z^2$ and found that gradient which was $\nabla f = i - 10yj - 14zk$.

Do I need the equation for the tangent plane of the paraboid? And are -1, 1, 1 the directional numbers for the plane $-x + y + z = 3$?
• Oct 3rd 2011, 09:21 AM
TheEmptySet
Re: Point on paraboloid at which the tangent plane is parallel to plane
Quote:

Originally Posted by deezy
Find pt on paraboloid $x = 5y^2 + 7z^2$, if it exists, at which the tangent plane is parallel to plane $-x + y + z = 3$.

Not completely sure how to approach this problem. I'm pretty sure it involves the gradient, so I set $f(x,y,z) = x - 5y^2 - 7z^2$ and found that gradient which was $\nabla f = i - 10yj - 14zk$.

Do I need the equation for the tangent plane of the paraboid? And are -1, 1, 1 the directional numbers for the plane $-x + y + z = 3$?

Two planes are parallel if their normal vectors are parallel.

The normal vector of the plane is $- \mathbf{i}+ \mathbf{j}+ \mathbf{k}$

or $\mathbf{i}- \mathbf{j}- \mathbf{k}$

Now just set this equal to the normal vector of the surface that you have found and find what the variables have to be.

$\nabla f = \mathbf{i} - 10y\mathbf{j} - 14z \mathbf{k}$
• Oct 3rd 2011, 06:52 PM
deezy
Re: Point on paraboloid at which the tangent plane is parallel to plane
Ok, here's what I've done:

$-10y = 1$
$-14z = 1$
$y = -1/10$
$z = -1/14$

I got confused at this point. I wasn't sure what I do with the x. $\nabla f = i -(10y)j-(14z)k$. There wasn't an x in the equation, and $1 \neq -1$, so does the point exist?

And is there a way to check, if I found a point, that it was a correct point?
• Oct 4th 2011, 10:59 AM
TheEmptySet
Re: Point on paraboloid at which the tangent plane is parallel to plane
Quote:

Originally Posted by deezy
Ok, here's what I've done:

$-10y = 1$
$-14z = 1$
$y = -1/10$
$z = -1/14$

I got confused at this point. I wasn't sure what I do with the x. $\nabla f = i -(10y)j-(14z)k$. There wasn't an x in the equation, and $1 \neq -1$, so does the point exist?

And is there a way to check, if I found a point, that it was a correct point?

You have forgotten that every plane has two normal vectors, as I pointed out in my original post. If you muliply the normal by -1 you get get another normal vector but with opposite direction.