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fresh_42

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First of all, ellipses and hyperbolas both have two foci, not one focus. The two foci are the two points in the definitions I provided, not two other points. So, if the two foci are

An ellipse (represented by the equation (x

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jbriggs444

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Are there equations? Of course. ##k = |AP| \cdot |BP|## is an equation. It could be written as ##k = \sqrt{(x-x_A)^2 + (y-y_A)^2} \cdot \sqrt{(x-x_B)^2 + (y-y_B)^2}##What I'm asking is if there are any shapes and equations for the sets of points such that || * |AP| is constant, |BP| / |AP| is constant, and/or |BP| / |BP| is constant.AP

Are you, perhaps, asking whether that equation could be simplified into some other form such as a polynomial equation in x and y using some constants which depend on k, ##\vec{A}## and ##\vec{B}##? It looks like squaring both sides would be a good start. I've not worked it out, but it looks like that particular result would be a degree 4 polynomial equation in x and y.

Are there shapes? Of course. You could plot the solution set for the above equation. I believe the solution set (for k>0, and ##\vec{A}## different from ##\vec{B}##) will always contain at least one and possibly two closed curves. Are you, perhaps, asking whether those curves are shapes which are well-known by other names? Beats me.

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Yeah, I suppose it was kind of a silly way for me to phrase my questions. Though I commend you for not only answering the questions I posed but also addressing and answering the questions I meant when I myself didn't fully realize that those alternate questions were far closer to expressing what I was actually curious about. Thank youAre there equations? Of course. ##k = |AP| \cdot |BP|## is an equation. It could be written as ##k = \sqrt{(x-x_A)^2 + (y-y_A)^2} \cdot \sqrt{(x-x_B)^2 + (y-y_B)^2}##

Are you, perhaps, asking whether that equation could be simplified into some other form such as a polynomial equation in x and y using some constants which depend on k, ##\vec{A}## and ##\vec{B}##? It looks like squaring both sides would be a good start. I've not worked it out, but it looks like that particular result would be a degree 4 polynomial equation in x and y.

Are there shapes? Of course. You could plot the solution set for the above equation. I believe the solution set (for k>0, and ##\vec{A}## different from ##\vec{B}##) will always contain at least one and possibly two closed curves. Are you, perhaps, asking whether those curves are shapes which are well-known by other names? Beats me.

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