Answer
1. $xy$-trace
The trace is a hyperbola in standard position, located in the $xy$-plane.
2. $yz$-trace
The trace is a hyperbola in standard position, located in the $yz$-plane.
3. $xz$-trace
The trace is a circle of radius $2$, located in the $xz$-plane.
Work Step by Step
1. $xy$-trace
We find the $xy$-trace by setting $z=0$, so we obtain ${\left( {\frac{x}{2}} \right)^2} - {y^2} = 1$.
Write ${\left( {\frac{x}{2}} \right)^2} - {\left( {\frac{y}{1}} \right)^2} = 1$. This is the equation of a hyperbola in standard position, located in the $xy$-plane.
2. $yz$-trace
We find the $yz$-trace by setting $x=0$, so we obtain $ - {y^2} + {\left( {\frac{z}{2}} \right)^2} = 1$.
Write ${\left( {\frac{z}{2}} \right)^2} - {\left( {\frac{y}{1}} \right)^2} = 1$. This is the equation of a hyperbola in standard position, located in the $yz$-plane.
3. $xz$-trace
We find the $xz$-trace by setting $y=0$, so we obtain ${\left( {\frac{x}{2}} \right)^2} + {\left( {\frac{z}{2}} \right)^2} = 1$.
Write ${x^2} + {z^2} = 4$. This is the equation of a circle of radius $2$, located in the $xz$-plane.
