Determining restrictions on an ellipse

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SUMMARY

The equation Ax² + By² + Cx = 1 defines an ellipse, where the conditions for the major axis to align with the y-axis are C = 0 and A > B. A common misconception is that A < B would yield the same result, which is incorrect. To validate these conditions, one can plot ellipses for both A > B and A < B scenarios, confirming that only A > B results in the major axis being vertical.

PREREQUISITES
  • Understanding of conic sections, specifically ellipses
  • Familiarity with the standard form of ellipse equations
  • Basic knowledge of graphing techniques
  • Ability to manipulate algebraic equations
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  • Explore the properties of ellipses in conic sections
  • Learn how to graph ellipses using software like Desmos or GeoGebra
  • Investigate the effects of varying coefficients A, B, and C on ellipse orientation
  • Study the general form of conic sections and their classifications
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Mathematicians, students studying geometry, and educators teaching conic sections will benefit from this discussion.

trigger352
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The equation [tex]Ax^2 + By^2 + Cx = 1[/tex] represents an ellipse. If [tex]A > 0[/tex] and [tex]B > 0[/tex], what conditions must be satisfied if the ellipse has it's major axis on the y-axis?

The answer is "[tex]C = 0[/tex] and [tex]A > B[/tex]"

When I first wrote this question I thought [tex]A > B[/tex] should have been [tex]A < B[/tex]. So how do I figure out what the restrictions are? Why is [tex]A < B[/tex] wrong?
 
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Plot an ellipse where A > B and see if the answer is correct... then test your way of A < B.

Jameson

BTW: Your answer is correct. If B > A , then the major axis will run along the y-axis.
 
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C different from zero would automatically shift the center of the ellipse from the center of coordinates...

Daniel.
 

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