Question inspired by reviewing conic sections

  • #1
Recently, in my calculus two class, we began going over conic sections. After reviewing the definitions of ellipses and hyperbolas - For two given points, the foci, an ellipse is the locus of points such that the sum of/difference between the distance to each focus is constant, respectively - I couldn't help but become curious: does a shape and/or equation for a set of points such that "the product or quotient of the distance to each focus is constant" instead of just the sum or difference? I posed this question to my math instructor too, but he didn't know. (I hope that this is the right section to post this in. It seemed most likely, but I couldn't tell with any certainty which sub-forum to post this on)
 

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  • #2
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Can you explain precisely what you mean? If I consider two points ##A,B## at a distance ##a,b## form a focus ##F##, i.e. ##a=|AF| , b=|BF|## and look out for a third point ##C## with distance ##c=|CF|## from the focus, such that ##ab=ac=bc##, then I get ##a=b=c## and end up with a circle or the set ##\{A,B,F\}##.
 
  • #3
Can you explain precisely what you mean? If I consider two points ##A,B## at a distance ##a,b## form a focus ##F##, i.e. ##a=|AF| , b=|BF|## and look out for a third point ##C## with distance ##c=|CF|## from the focus, such that ##ab=ac=bc##, then I get ##a=b=c## and end up with a circle or the set ##\{A,B,F\}##.
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 A and B, and the distances between those foci and any given point P are |AP| and |BP|:
An ellipse (represented by the equation (x2/a2) + (y2/b2) = 1, or (y2/a2) + (x2/b2) = 1 with a>b) consists of all points P such that |AP| + |BP| is constant and a hyperbola (represented by the equation (x2/a2) - (y2/b2) = 1, or (y2/a2) - (x2/b2) = 1) consists of all points P such that the absolute value of |AP| - |BP| is constant. What I'm asking is if there are any shapes and equations for the sets of points such that |AP| * |BP| is constant, |AP| / |BP| is constant, and/or |BP| / |AP| is constant.
 
  • #4
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What I'm asking is if there are any shapes and equations for the sets of points such that |AP| * |BP| is constant, |AP| / |BP| is constant, and/or |BP| / |AP| is constant.
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}##

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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  • #5
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}##

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.
:-pYeah, 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 you
 
  • #6
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If you make a contour plot of ##z = \sqrt{(x-x_A)^2 + (y-y_A)^2} \cdot \sqrt{(x-x_B)^2 + (y-y_B)^2}##, you should be able to see what the level curves look like.
 

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