APSC 172 Lecture Notes - Lecture 4: Quadratic Equation, Tangent Space

• School: Queen's University
• Department: Applied Science
• Course: APSC 172
• Professor: Ping Li

Document Summary

Find all tangent planes to the elliptic paraboloid that contain the line x z = 4 x2 2y2 t. [ans: z = 7 + 2x 4y, z = 7 10x/3 + 4y/3] There are many ways to tackle this, but here we will take as our unknown the coordinates (a, b, c) of the point of contact of the plane with the surface. In fact since this point will be on the surface, it will have the algebraic form (a, b, 4 a2 2b2). First we find the tangent plane to the surface at this point. The partial derivatives of z are: z x. The tangent plane has the form: z y. 2 b a y x c z ax by. Can you see that there ought to be (at least) two, one on each side of the hill. Think of a pup-tent, the line being the ridge pole.