Answer
Elliptic cone, with x as the axis.
Work Step by Step
Here, we have: $x^2=y^2 +4z^2$
Comparing the above form with the equation $\displaystyle \frac{z^{2}}{c^{2}}=\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}$, we find:
In this case, we see that the horizontal traces are ellipses (z is the axis).
Vertical traces in the planes $x=k$ and $y=k$ are hyperbolas when $k\neq 0$ but are pairs of lines when $k=0$.
When $x=k$ (axis is the x-axis), traces in the planes $y=k$ and $z=k$ are hyperbolas.
The traces are lines, so we have an elliptic cone.
