- #1
Alw
- 8
- 0
Homework Statement
(a) Find two linear equations in the variables x,y,z such that their solution set is the tangent line to the graph of f(x,y) = [tex]x^{2}[/tex] - [tex]y^{2}[/tex] at (2,1,3) which is parallel to the x,z plane. (b) Represent the line of (a) by parametric equations.
Homework Equations
[tex]\nabla[/tex]f(x,y,z) = f([tex]f_{x}[/tex], [tex]f_{y}[/tex], [tex]f_{z}[/tex])
The tangent plane to the surface at a point:
0 = [tex]f_{x}[/tex](a,b,c)(x-a) + [tex]f_{y}[/tex](a,b,c)(y-b) + [tex]f_{z}[/tex](a,b,c)(z-c)
The Attempt at a Solution
I know how to solve for the plane (line in a function defined in two variables) tangent to the graph, and how to represent it parametrically, so I started with part b.
[tex]\nabla[/tex]f = (2x, -2y)
plugging into the tangent equation the point (2,1,3) for (a,b,c) and the components of the gradient gives
0 = 4(x-2) - 2(y-1)
and represented parametrically:
x = 2 + 4t, y = 1 - 2t (solution to part b)
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Looking again at part (a), he wants two linear equations such that their solution set is 0 = 4(x-2) - 2(y-1). I've been through the sections a few times and I haven't found any examples asking for this, and the other students in my class were struggling with this part too. Any ideas?