Question inspired by reviewing conic sections

[ScientificMind]

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)

 

[fresh_42]

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 $\{A,B,F\}$.

 

[ScientificMind]

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.

 

[jbriggs444]

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.

 

[ScientificMind]

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.

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 you

 

[FactChecker]

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.