Linear Program with "Same Direction" Constraint

Question

I have an optimization problem $$ \min_{x, y, t} \left\{ f(x, y) : g(x, y) \ge 0, ~ x = t y, ~t \ge 0 \right\} $$ where $x, y \in \mathbb R^n$ and $t$ is some non-negative scalar. In other words, $x$ and $y$ are constrained to have equal direction. Is this constraint $x = t y$ inherently quadratic, or does anyone have some trick to express it in a linear form?