Plotting the points that fulfill (x^2)+4(y^2)+(z^2)=4.

We could solve for z(x,y), and then the 2 parameters to use to plot the surface are x and y:

({{({{z=±(Sqrt[4-(x^2)-4(y^2)])}})}})

P1=ParametricPlot3D[{x,y,Sqrt[4*(1-x^2/4-y^2)]},{x,-2,2},{y,-1,1}]
P2=ParametricPlot3D[{x,y,-Sqrt[4*(1-x^2/4-y^2)]},{x,-2,2},{y,-1,1}]
Show[P1,P2,PlotRange->All,AspectRatio->1]

An alternative approach to plotting (x^2)+4(y^2)+(z^2)=4:

Rearrange this as   (x^2)+(z^2)=4(1-(y^2)).

We recognize (x^2)+(z^2)=(r^2) as the equation of a circle of radius (r^2)(y)=4(1-(y^2)) .

To draw a circle, we could use a parameter θ which runs from 0 to 2π, together with a parameter that represents the radius, r and write:

({{({{x(r,θ)=r cosθ}})
z(r,θ)=r sinθ}})

To get y(r,θ), we solve the equation for r(y) for y:

({{({{y=±(Sqrt[1-(r^2)/4])}})}})

This only works if |r|≤2.

P3=ParametricPlot3D[
{r Cos[θ],Sqrt[1-1/4 r^2],r Sin[θ]},
{θ, 0, 2 π},{r,0,2}
]
P4=ParametricPlot3D[
{u Cos[v],-Sqrt[1-1/4 u^2],u Sin[v]},
{v, 0, 2 π},{u,0,2}

]
Show[P3,P4, PlotRange->All,AspectRatio->1]