Points on the curve of intersection

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SUMMARY

The discussion focuses on finding the points on the curve of intersection defined by the equations \(x^2 + (y-1)^2 + (z-1)^2 = 1\) and \(2x + y + 2z = 4\). The user seeks to determine the nearest and furthest points from the origin, along with the minimum and maximum distances. The suggested approach involves solving for one variable, specifically \(y = 4 - 2x - 2z\), and substituting it into the other equation. The problem is identified as suitable for Lagrange multipliers to optimize the distance function.

PREREQUISITES
  • Understanding of 3D geometry and curves
  • Familiarity with the equations of spheres and planes
  • Knowledge of optimization techniques, specifically Lagrange multipliers
  • Ability to manipulate and solve multivariable equations
NEXT STEPS
  • Study the method of Lagrange multipliers for optimization problems
  • Learn how to derive the distance formula in three-dimensional space
  • Explore the geometric interpretation of curves of intersection
  • Practice solving systems of equations involving spheres and planes
USEFUL FOR

Students in calculus or multivariable calculus courses, mathematicians interested in optimization problems, and anyone studying geometric intersections in three-dimensional space.

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hi i need to find the points on the curve of intersection of x^2+(y-1)^2+(z-1)^2 = 1 and 2x + y + 2z = 4 which are nearest and furthest furthest from the origin. Also the min and max distances. I'm not looking for you guys to do this four me, I'm kind of lost and don't know where to go. I think I'm supposed to solve one of the equations for a variable such as y = 4 - 2x - 2z and plug it into the other equation to get the curve of intersection but i;m not sure what to do from there and if that's even correct. I think my teacher also said something about setting one of the equations = to z cause tahts going to be the maximum.
 
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This sounds like a Lagrange multipliers problem!
 

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