# Plot ellipses in a exponential curve

I am trying to produce a single plot, where the ellipses fits neatly within the given curve function, and with each intersecting the curve at exactly 4 points.

• The ellipses is $$x^2/a^2+y^2/b^2=1$$, where $$a=0.5, 0.6, 0.7, ..., 2.0$$, where $$a>0$$,$$b>0$$ and $$πab=5$$. Hint: use Table to create all of them in one command.

• The curve $$|y|=5/(2π|x|)$$, drawn in black. Be sure that the curve appears, and draw the single curve on top of the ellipses so that it is clearly visible.

What I tried to do was

Clear[x, y]
ellipse = x^2/a^2 + y^2/b^2 == 1;
curve = Abs[y] == 5/(2*Pi*Abs[x]);
ell = Table[ellipse, {a, 0.5, 2.0, 0.5}, {b, 5/Pi*a}];
curvep = ContourPlot[Evaluate[curve], {x, -7, 7}, {y, -7, 7},
ContourStyle -> Black];
ellipsep =
ContourPlot[Evaluate[ell], {x, -7, 7}, {y, -7, 7},
ContourStyle -> Rainbow];
Show[ellipsep, curvep]


but for some reason, my ellipses are not fitting inside the curve function and rather it is outside. • With[{ellipses = Table[Circle[{0, 0}, {a, 5/(a \[Pi])}], {a, 0.5, 2.0, 0.1}]}, ContourPlot[Abs[y] == 5/(2 Pi Abs[x]), {x, -7, 7}, {y, -7, 7}, Epilog -> {Opacity[.5], ellipses}, ContourStyle -> Thick] ] Oct 10, 2020 at 22:38
• Can you use a different command without "With". Oct 10, 2020 at 22:45
• Why is the range for a {0.5,2.0,0.1}? where did 0.1 come from? Oct 10, 2020 at 22:48
• The requirements for ellipses are “where 𝑎=0.5,0.6,0.7,...,2.0”, which tells us that the increment between 0.5 and 0.6, and between 0.6 and 0.7, etc. is 0.1. If you use 0.5, then 𝑎=0.5,1.0,1.5,2.0 which is not what’s required. Oct 10, 2020 at 23:07

In the equations of ellipse, replace b by 5/(Pi*a) so just one parametric say a, then it is easy to create a table ell satisfy Pi*a*b=5

Clear["*"];
Clear[x, y]
ellipse = x^2/a^2 + y^2/b^2 == 1 /. b -> 5/(Pi*a);
curve = Abs[y] == 5/(2*Pi*Abs[x]);
ell = Table[ellipse, {a, 0.5, 2.0, 0.5}];
curvep = ContourPlot[Evaluate[curve], {x, -7, 7}, {y, -7, 7},
ContourStyle -> Black];
ellipsep =
ContourPlot[Evaluate[ell], {x, -7, 7}, {y, -7, 7}];
Show[ellipsep, curvep]
` 