Finding Parametric Equation of Tangent Line to Intersection of Surfaces?

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SUMMARY

The discussion focuses on finding the parametric equation of the tangent line to the intersection of the paraboloid defined by the equation z = x² + y² and the ellipsoid given by 4x² + y² + z² = 9 at the point (-1, 1, 2). The key method involves calculating the gradients of both surfaces at the specified point and then taking the cross product of these gradients to determine the direction vector of the tangent line. This approach eliminates the need to derive the full equation of the intersection curve, simplifying the problem significantly.

PREREQUISITES
  • Understanding of parametric equations in three-dimensional space
  • Knowledge of gradients and their geometric interpretations
  • Familiarity with cross products of vectors
  • Basic concepts of paraboloids and ellipsoids in multivariable calculus
NEXT STEPS
  • Study the method of finding gradients for multivariable functions
  • Learn how to compute cross products of vectors in three-dimensional space
  • Explore the geometric interpretations of tangent lines to curves of intersection
  • Investigate the properties of paraboloids and ellipsoids in calculus
USEFUL FOR

Students and educators in multivariable calculus, particularly those focusing on surface intersections and tangent line calculations. This discussion is beneficial for anyone seeking to enhance their understanding of vector calculus concepts.

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Homework Statement


Hi, I need help with the following. I'm asked to find the parametric equation of the tangent line to the curve of the interrsection of the paraboloid z = x^2 + y^2 and the ellipsoid 4x^2+y^2+z^2 = 9 at the point (-1,1,2).


Homework Equations


I think I'm asked to find the gradient of the curves of the intersection, and then I know the vector with the direction of the line eof intersection, and then I can plug it back into find the parametric equation of the line.


The Attempt at a Solution


I guess I am trying to find the equation of the curve of intersection first? So, would I set the two equations equal so that I would get 4x^2+y^2+(x^2+y^2)^2 = 9? But then z has gone away? I guess I'm pretty clueless as to how to attempt to solve this problem?
Or if I'm right that the intersection is an ellipse, but how am I supposed to find the equation of the ellipse? Can anyone give me any pointers or hints?
 
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You are only asked to find the tangent line at one particular point, so you don't need to find the equation of the complete intersection curve and then find its tangent. Doing all that would give you the answer, though.

Think about the tangent planes to the two surfaces at the point (-1,1,2).
 
Was going to post. Figured it out. You only need to take the cross product of the gradient to the two curves.
 

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