Basic question: Is it possible for the equation of a circle to have an xy term?

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Discussion Overview

The discussion revolves around whether the equation of a circle can include an xy term. Participants explore the implications of such a term in the context of conic sections and the conditions under which a circle can be represented by a given equation.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes the equation x^2 + y^2 - 2hx - 2ky + 2mxy + r = 0 as a candidate for a circle.
  • Another participant argues that the presence of the xy term indicates a rotated conic section, suggesting that circles do not have cross terms when expressed in standard form.
  • A different viewpoint states that the presence of "xy" indicates a rotation of the axes, but asserts that a circle remains unchanged under rotation.
  • Some participants suggest that if m = 0, the equation may not include the xy term, leading to conditions under which the equation could represent a circle.
  • Further contributions indicate that for the equation to represent a circle, specific conditions must be met, such as h^2 + k^2 < r.
  • One participant emphasizes that the correct interpretation of the original question leads to the conclusion that the equation cannot have an xy term, while also noting that under certain conditions, it may be possible for the equation to represent a circle.

Areas of Agreement / Disagreement

Participants do not reach a consensus. There are competing views regarding the implications of the xy term in the equation of a circle, with some asserting it cannot exist while others propose conditions under which it might.

Contextual Notes

Participants express varying assumptions about the parameters involved, particularly regarding the values of m, h, k, and r, which influence the interpretation of the equation.

seeker101
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Can this be an equation of a circle x^2+y^2-2hx-2ky+2mxy+r=0
 
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No. The presence of xy denotes a term of the form (Ax + By)2, which does not form any part of the general equation of a circle. It may, however, be a conic of some sort that is simply rotated. You can get cross terms by applying a rotation change of coordinates (x, y) -> (x', y') to the general equation of a conic, x' = Cx - Sy, y' = Sx + Cy, where S= sin(t) and C = cos(t) where t is the angle of rotation. In the case of a circle, the cross terms cancel.
You can find the general equation of a conic section here.
 
The presence of "xy" indicates rotation of the axes. And a circle looks exactly the same no matter how it is rotated!
 
Perhaps, if m = 0.
 
I guess it can when m = 0, r < k2 + h2
 
The question in the title was "Is it possible for the equation of a circle to have an xy term" to which the correct answer is "No".

The question in the text was "Can this be the equation of a circle x^2+ y^3- 2hx- 2kx+ 2mxy= r" to which the answer is "Yes". It is possible with m= 0 (so the equation does not have a an xy term) and h^2+ y^2&lt; r, NOT r&lt; h^2+ k^2. If m= 0 and h^2+ k^2= r the graph is the single point (h, k). If m= 0 h^2+ k^2&gt; r there are no (x, y) points satifying the equation.
 

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