Answer
Hyperboloid of one sheet; the trace is a hyperbola in the $ xz $-plane.
Work Step by Step
The equation
$$
\left(\frac{x}{3}\right)^{2}+ \left(\frac{y}{5}\right)^{2} -5z^2=1
$$
can rewritten as follows
$$
\left(\frac{x}{3}\right)^{2}+ \left(\frac{y}{5}\right)^{2} -\left(\frac{z}{\sqrt 5}\right)^{2}=1
$$
which is a hyperboloid of one sheet.
(See equations on page 691.)
To find the trace with the plane $ y=1$, we have
$$
\left(\frac{x}{3}\right)^{2}-\left(\frac{z}{\sqrt 5}\right)^{2}=\frac{24}{25}
$$
which is a hyperbola in the $ xz $-plane.
