2Given the cone 4x² +16y² = 2², find aparameterization for the cone,Find a normal vector and tangent planeat (0₁-1,4),(u, v).

Given equation of cone is Let,Let,Now Now

Answered: 2 Given the cone 4x² +16y² = 2², find a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

2
Given the cone 4x² +16y² = 2², find a
parameterization for the cone,
Find a normal vector and tangent plane
at (0₁-1,4),
(u, v).

Expert Solution

Step 1: Given information and parametric form of cone

Given equation of cone is $4 x squared plus 16 y squared equals z squared$

Let, $x equals u space a n d space y equals v space t h e n space z equals square root of open parentheses 4 u squared plus 16 v squared close parentheses end root$

Let, $phi open parentheses u comma v close parentheses equals open parentheses u comma v comma square root of open parentheses 4 u squared plus 16 v squared close parentheses end root space close parentheses$

Now $fraction numerator partial differential phi over denominator partial differential u end fraction equals open parentheses 1 comma 0 comma fraction numerator 4 u over denominator square root of open parentheses 4 u squared plus 16 v squared close parentheses end root end fraction close parentheses space a n d space fraction numerator partial differential phi over denominator partial differential v end fraction equals open parentheses 0 comma 1 comma fraction numerator 16 v over denominator square root of open parentheses 4 u squared plus 16 v squared close parentheses end root end fraction close parentheses$

Now $fraction numerator partial differential phi over denominator partial differential u end fraction cross times fraction numerator partial differential phi over denominator partial differential v end fraction equals open parentheses negative fraction numerator 4 u over denominator square root of open parentheses 4 u squared plus 16 v squared close parentheses end root end fraction comma negative fraction numerator 16 v over denominator square root of open parentheses 4 u squared plus 16 v squared close parentheses end root end fraction comma 1 close parentheses$