(* Ellipsoid*)
TraditionalForm[x^2/a^2 + y^2/b^2 + z^2/c^2==1]
Manipulate[
ParametricPlot3D[{a*Cos[θ]*Sin[ϕ],b*Sin[θ]*Sin[ϕ],c*Cos[ϕ]},{θ,0,2π},{ϕ,0,2π},AxesOrigin->{0,0,0},PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},0.01,5}]

(*Cone*)
TraditionalForm[z^2/c^2==x^2/a^2 + y^2/b^2 ]
Manipulate[
ParametricPlot3D[{a*z*Cos[θ],b*z*Sin[θ],c*z},{θ,0,2π},{z,-3,3},AxesOrigin->{0,0,0},PlotRange->{{-7,7},{-7,7},{-7,7}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},0.01,5}]

(*Elliptic Paraboloid*)
TraditionalForm[z/c==x^2/a^2 + y^2/b^2 ]
Manipulate[
ParametricPlot3D[{a*z*Cos[θ],b*z*Sin[θ],c*z^2},{θ,0,2π},{z,0,3},AxesOrigin->{0,0,0},PlotRange->{{-7,7},{-7,7},{-7,7}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},-5,5}]

(*Hyperboloid of One Sheet*)
TraditionalForm[x^2/a^2 + y^2/b^2 -z^2/c^2==1]
Manipulate[
ContourPlot3D[x^2/a^2 + y^2/b^2 -z^2/c^2==1,{x,-5,5},{y,-5,5},{z,-5,5},AxesOrigin->{0,0,0},PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},0.01,5}]

(*Hyperboloid of Two Sheets*)
TraditionalForm[-x^2/a^2 - y^2/b^2 +z^2/c^2==1]
Manipulate[
ContourPlot3D[x^2/a^2 + y^2/b^2 -z^2/c^2==-1,{x,-5,5},{y,-5,5},{z,-5,5},AxesOrigin->{0,0,0},PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},0.01,5}]

(*Hyperbolic Paraboloid*)

TraditionalForm[z/c==x^2/a^2 -y^2/b^2 ]
Manipulate[
ContourPlot3D[z==c(x^2/a^2 -y^2/b^2),{x,-5,5},{y,-5,5},{z,-5,5},AxesOrigin->{0,0,0},PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False],
{{a,1},0.01,5},{{b,1},0.01,5},{{c,1},-5,5}]

(*Connection Between Hyperboloid of One Sheet and of Two Sheets*)

TraditionalForm[x^2+ y^2 -z^2==-k]
Manipulate[
ContourPlot3D[x^2+ y^2 -z^2==-k,{x,-5,5},{y,-5,5},{z,-5,5},AxesOrigin->{0,0,0},PlotRange->{{-5,5},{-5,5},{-5,5}},Boxed->False],
{{k,-1},-1,1}]