Thread: Point on paraboloid at which the tangent plane is parallel to plane

Find pt on paraboloid , if it exists, at which the tangent plane is parallel to plane .

Not completely sure how to approach this problem. I'm pretty sure it involves the gradient, so I set and found that gradient which was .

Do I need the equation for the tangent plane of the paraboid? And are -1, 1, 1 the directional numbers for the plane ?

Originally Posted by deezy

Find pt on paraboloid , if it exists, at which the tangent plane is parallel to plane .

Not completely sure how to approach this problem. I'm pretty sure it involves the gradient, so I set and found that gradient which was .

Do I need the equation for the tangent plane of the paraboid? And are -1, 1, 1 the directional numbers for the plane ?

Two planes are parallel if their normal vectors are parallel.

The normal vector of the plane is

or

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

Ok, here's what I've done:






I got confused at this point. I wasn't sure what I do with the x. . There wasn't an x in the equation, and , so does the point exist?

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

Originally Posted by deezy

Ok, here's what I've done:






I got confused at this point. I wasn't sure what I do with the x. . There wasn't an x in the equation, and , 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.