consider $4x^2 + y^2 = 4$. What I have if $x=0$?

Question

I need to graph the following surface: $4x^2 + y^2 = 4$ which is an ellipse, so canonical equation is $x^2 + (y^2)/4 = 1$. Then, to graph this I can say:

If $x=0$:

$y^2 = 4$

so $y = 2$ or $y = -2$

But what I have in the plane? two parallels lines? because I just see two points at that coords in the following graph:

Answer 1

If you are graphing in xyz space the surface is a tube with a constant x/y cross section being an ellipse for any $z = k$.

If you take the y/z cross section at $x = 0$ this is cutting the tube lengthwise in half and seeing the cut lines. The result is two parallel lines.

That's exactly what you should expect.

If you are graphing only on the x-y plane the this is an ellipse. Setting $x = 0$ gives you the two points on the ellipse where $x =0$. That there are two points and they are $(0,2)$ and $(0,-2)$ should not surprise any one.

Answer 2

You get parallel lines when $y= -2$ or $ y = $ any constant $c$ FOR EVERY $x$. Please be careful, in the context or your question its only the case that $y= 2$ or $y = -2$ when $x=0$ not at any other points.