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Mathematical Expression Editor
A plane perpendicular to the -plane contains the point on the paraboloid . The line
tangent to the cross-section of the paraboloid created by this plane is parallel to the
-plane at this point. Find an equation of the plane.
Let be the normal vector for the plane in question.
We know that because the plane is perpendicular to the -plane.
The paraboloid can be thought of as the level surface of some new function of
several variables:
The gradient vector is normal to level surfaces.
Since line tangent to the cross-section of the paraboloid created by this plane is
parallel to the -plane at , the tangent vector of the cross-section is parallel to
.
Moreover, since this vector is parallel to the -plane,
So .
This means that is parallel to .
Set . Now since we know a point on the plane, we can find the plane.