What is the equation for level curves in an elliptical shape?

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SUMMARY

The discussion centers on deriving the equation for level curves of the function z = (5x² + y²)^(1/2) - 2x. Participants aim to demonstrate that these curves are elliptical by reformulating the equation into the standard form of an ellipse, Ax² + By² = R². The initial transformation leads to the equation z² + 4xz = x² + y², which participants suggest can be analyzed without isolating z. They recommend using the general conic equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 for further analysis.

PREREQUISITES
  • Understanding of conic sections, specifically ellipses
  • Familiarity with algebraic manipulation and equation transformation
  • Knowledge of level curves in multivariable functions
  • Basic calculus concepts related to functions of two variables
NEXT STEPS
  • Study the properties of ellipses and their standard equations
  • Learn how to manipulate equations of conics, focusing on the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0
  • Explore the concept of level curves in multivariable calculus
  • Practice algebraic techniques for isolating variables in complex equations
USEFUL FOR

Students studying calculus, particularly those focusing on multivariable functions and conic sections, as well as educators looking for examples of level curves and their properties.

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Homework Statement



Describe the shape of each level curve for the following function:

z= (5x^2+y^2)^.5-2x


Homework Equations



I would like to prove that the curves are elliptical by setting z as a constnat and algebraically putting the equation in standard for for an ellipse Ax^2+By^2=R^2


The Attempt at a Solution



After squaring both sides, I get to:

z^2+4xz=x^2+y^2

I do not know how to isolate the z on one side from there. Any suggestions? I feel like this is a really obvious algebra trick that I am forgetting.

Thank you very much for any help.
 
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Two suggestions:

- think of z as a number (that is arbitrary). No reason to isolate it

- Use the equation of the general conic Ax2+Bxy+Cy2+Dx+Ey+F=0
 

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