Plot ellipses in a exponential curve

Question

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.

Answer 1

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]