Hello math help forums! This is my first post on these forums and I hope to come back here often. Go calc!

Okay, so I just finished a problem on my homework, but for some reason the logic I used in completing this problem seems odd. I don't know why though.

Here is the problem:

Find a point on the surface of at which the tangent plane is perpendicular to the line . Find the equation of the tangent plane at that point.

First, I know that a plane in space is defined by normal vector to the plane and a point on that plane. Because the given line is perpendicular to the tangent plane, I used the direction numbers of the line to form a scalar equation for the tangent plane:

Where is the point on the surface at which the tangent plane is perpendicular to the line.

The equation of the tangent plane is also given by:

Thus,

Since both equations equal 0, and because the point is the same for each equation, I set the equations equal to each other. For some reason, I feel like I am doing something wrong at this point... I don't know why.

The z terms cancel out, and I am left with:

Thus,

and

I then plugged these points into the original equation for the surface to obtain

Does this logic seem right?

Thank you for your help! It is greatly appreciated.

Sincerely,

Opiaboy