(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The ellipsoid 4x^2+2y^2+z^2=16 intersects the plane y=2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2)

2. Relevant equations

sin(t)^2 + cos(t)^2 = 1

3. The attempt at a solution

After plugging 2 in for y, I get 4x^2 + z^2 = 8

thus x^2/2 + z^2/8 = 1

if sin(t)^2 + cos(t)^2=1

then x^2/2=sin(t)^2

and x = root(2)sint(t)

z would = 2root(2)cos(t)

and thus the curve of

x=root(2)sint(t), y=2, z=2root(2)cos(t) would be the intersect of the plane and the ellipsoid.

to get a tangent of the line, it would be r'(t)/|r'(t)

at the point x = 1 = root(2)sint(t) -> t = 45degrees meaning that all cos(t) and sin(t) would be root(2)/2

however for r'(t), I get root(2)cos(t)i + 0j + -2root(2)cos(t) and plugging in root(2)/2, will yield a -1.

Finding the magnitude of the vector always gets me a root(5)

However the answer should be x = 1+t, y=2, z=2-2t

am I making this too complicated or do I not get how to parametrize the intersection of a surface and a plane?

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# Parametric equations for Tangent line of an ellipse

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