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Parametric equations for Tangent line of an ellipse 
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#1
Oct810, 03:31 AM

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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=22t am I making this too complicated or do I not get how to parametrize the intersection of a surface and a plane? 


#2
Oct810, 11:16 AM

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You are being much too complicated!
First the problem asked for parametric equations for the tangent line. It is not necessary to get parametric equations for the ellipse,. Yes, with y= 2, your equation reduces to [itex]4x^2+ z^2= 8[/itex]. From that you can calculate immediately that [itex]8x + 2z dz/dx= 0[/itex] so that [itex]dz/dx= 4x/z[/itex]. In particular, at x= 1, z= 2, [itex]dz/dx= 4(1)/2= 2[/itex]. The tangent line through that point, in the y= 2 plane, is z= 2x+ 4. Now, to get parametric equations of the line in three dimensions, take x= t as parameter: x= t, y= 2, z= 2t+ 4. 


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