Note that this requires Mathematica to evaluate the cells, and I plan on using this in class. It will not be much use if you don’t have Mathematica installed on your computer, or if you have not used Mathematica before.

Evaluation > Evaluate Notebook

Ellipses: ((((x-h))^2)/(a^2))+((((y-k))^2)/(b^2))=1

((x^2)/(a^2))+((y^2)/(1^2))=1

w=8;
Clear[eplot,efocus1,efocus2,erectangle]
Do[
eplot[a]=ContourPlot[x^2/a^2+y^2==1,{x,-w,w},{y,-w,w},AspectRatio->1,ContourStyle->Thick];
center=Graphics[{Blue,Disk[{0,0},0.15]}];
c=Sqrt[a^2-1^2];
efocus1[a]=Graphics[{Red,Disk[{c,0},0.15]}];
efocus2[a]=Graphics[{Red,Disk[{-c,0},0.15]}];
erectangle[a]=Graphics[{Orange,Opacity[0.5],Rectangle[{-a,-1},{a,1}]}]
,{a,1,5,1/50}];

Manipulate[
Show[erectangle[a],eplot[a],center,efocus1[a],efocus2[a],PlotRange->{{-w,w},{-w,w}},AspectRatio->1,Frame->True,ImageSize->{600,Automatic}]
,{a,1,5,1/50}]

((x^2)/(1^2))+((y^2)/(b^2))=1

Clear[eplotb,efocus1b,efocus2b,erectangleb]
Do[
eplotb[b]=ContourPlot[x^2+y^2/b^2==1,{x,-w,w},{y,-w,w},AspectRatio->1,ContourStyle->Thick];
center=Graphics[{Blue,Disk[{0,0},0.15]}];
c=Sqrt[b^2-1^2];
efocus1b[b]=Graphics[{Red,Disk[{0,c},0.15]}];
efocus2b[b]=Graphics[{Red,Disk[{0,-c},0.15]}];
erectangleb[b]=Graphics[{Orange,Opacity[0.5],Rectangle[{-1,-b},{1,b}]}]
,{b,1,5,1/50}];

Manipulate[
Show[erectangleb[b],eplotb[b],center,efocus1b[b],efocus2b[b],PlotRange->{{-w,w},{-w,w}},AspectRatio->1,Frame->True,ImageSize->{600,Automatic}]
,{b,1,5,1/50}]

((x^2)/(a^2))+((y^2)/(b^2))=1

Clear[eplotab,efocus1ab,efocus2ab,erectangleab]
Do[
eplotab[a,b]=ContourPlot[x^2/a^2+y^2/b^2==1,{x,-w,w},{y,-w,w},AspectRatio->1,ContourStyle->Thick];
center=Graphics[{Blue,Disk[{0,0},0.15]}];
If[b>a,
{
c=Sqrt[b^2-a^2];
efocus1ab[a,b]=Graphics[{Red,Disk[{0,c},0.15]}];
efocus2ab[a,b]=Graphics[{Red,Disk[{0,-c},0.15]}];
},
{
c=Sqrt[a^2-b^2];
efocus1ab[a,b]=Graphics[{Red,Disk[{c,0},0.15]}];
efocus2ab[a,b]=Graphics[{Red,Disk[{-c,0},0.15]}];
}
];
erectangleab[a,b]=Graphics[{Orange,Opacity[0.5],Rectangle[{-a,-b},{a,b}]}]
,{a,1,5,1/10},{b,1,5,1/10}];

Manipulate[
Show[erectangleab[a,b],eplotab[a,b],center,efocus1ab[a,b],efocus2ab[a,b],PlotRange->{{-w,w},{-w,w}},AspectRatio->1,Frame->True,ImageSize->{600,Automatic}]
,{a,1,5,1/10},{b,1,5,1/10}]

Hyperbolas of Form:((((y-k))^2)/(b^2))-((((x-h))^2)/(a^2))=1

((y^2)/(b^2))-((x^2)/(a^2))=1

Clear[hplotabud,hfocus1abud,hfocus2abud,hrectangleabud,aplot1abud,aplot2abud]
Do[
hplotabud[a,b]=ContourPlot[y^2/b^2-x^2/(a^2)==1,{x,-w,w},{y,-w,w},AspectRatio->1,ContourStyle->Thick];
center=Graphics[{Blue,Disk[{0,0},0.15]}];
c=Sqrt[a^2+b^2];
hfocus1abud[a,b]=Graphics[{Red,Disk[{0,b+c},0.15]}];
hfocus2abud[a,b]=Graphics[{Red,Disk[{0,-b-c},0.15]}];
hrectangleabud[a,b]=Graphics[{Orange,Opacity[0.5],Rectangle[{-a,-b},{a,b}]}];
aplot1abud[a,b]=ParametricPlot[{(1-t)(-a)+t(a),(1-t)(-b)+t(b)},{t,-4,4},PlotStyle->{Dashed,Thick}];
aplot2abud[a,b]=ParametricPlot[{(1-t)(a)+t(-a),(1-t)(-b)+t(b)},{t,-4,4},PlotStyle->{Dashed,Thick}];
,{a,1,3,1/10},{b,1,3,1/10}];

Manipulate[
Show[hrectangleabud[a,b],hplotabud[a,b],center,hfocus1abud[a,b],hfocus2abud[a,b],aplot1abud[a,b],aplot2abud[a,b],PlotRange->{{-w,w},{-w,w}},AspectRatio->1,Frame->True,ImageSize->{600,Automatic}]
,{a,1,3,1/10},{b,1,3,1/10}]

Hyperbolas of Form: ((((x-h))^2)/(a^2))-((((y-k))^2)/(b^2))=1

((x^2)/(a^2))-((y^2)/(b^2))=1

Clear[hplotab,hfocus1ab,hfocus2ab,hrectangleab,aplot1ab,aplot2ab]
Do[
hplotab[a,b]=ContourPlot[x^2/a^2-y^2/b^2==1,{x,-w,w},{y,-w,w},AspectRatio->1,ContourStyle->Thick];
center=Graphics[{Blue,Disk[{0,0},0.15]}];
c=Sqrt[a^2+b^2];
hfocus1ab[a,b]=Graphics[{Red,Disk[{a+c,0},0.15]}];
hfocus2ab[a,b]=Graphics[{Red,Disk[{-a-c,0},0.15]}];
hrectangleab[a,b]=Graphics[{Orange,Opacity[0.5],Rectangle[{-a,-b},{a,b}]}];
aplot1ab[a,b]=ParametricPlot[{(1-t)(-a)+t(a),(1-t)(-b)+t(b)},{t,-4,4},PlotStyle->{Dashed,Thick}];
aplot2ab[a,b]=ParametricPlot[{(1-t)(a)+t(-a),(1-t)(-b)+t(b)},{t,-4,4},PlotStyle->{Dashed,Thick}];
,{a,1,3,1/10},{b,1,3,1/10}];

Manipulate[
Show[hrectangleab[a,b],hplotab[a,b],center,hfocus1ab[a,b],hfocus2ab[a,b],aplot1ab[a,b],aplot2ab[a,b],PlotRange->{{-w,w},{-w,w}},AspectRatio->1,Frame->True,ImageSize->{600,Automatic}]
,{a,1,3,1/10},{b,1,3,1/10}]