I need help plotting a curve on a graph where the distance from focus1 is always the same ratio to the distance from focus2.
For instance, lets assume focus1 is -5 along the x axis, and focus2 is +5 along the x axis.
If we plot a point at +4, this point is 9 units from focus1, and 1 unit from focus2. It's a 9 to 1 ratio, or 90% along the x axis from focus1 to focus2. I would like to plot everywhere else on the graph that the ratio is 9 to 1. I imagine this will make some sort of ellipse that spills out past focus2.
If we plot a point at 0, this point is 5 units from focus1, and 5 units from focus2. It's a 1 to 1 ratio, or 50% along the x axis from focus1 to focus2. Plotting a curve where every point is equal distance from both foci will make a straight line straight up the y axis.
If it helps to describe what I'm doing in greater detail, just let me know and I would be happy to oblige.
Can anyone help me with this?
Let the $\big(\pm a,0\big)$ be two foci. Then $~\dfrac{\sqrt{(x-a)^2+y^2}}{\sqrt{(x+a)^2+y^2}}=k.~$ For $k=1$, we have the segment's
perpendicular bisector; in this case, OY : $x=0$. Otherwise, after squaring and multiplying with the denominator, we get a simple circle. After all, a straight line can be interpreted as a circle of infinite radius.