  This note proves two theorems regarding Fermat-type equation $x^r + y^r =
dz^p$ where $r \geq 5$ is a prime. Our main result shows that, for infinitely
many integers~$d$, the previous equation has no non-trivial primitive solutions
such that $2 \mid x+y$ or $r \mid x+y$, for a set of exponents $p$ of positive
density. We use the modular method with a symplectic argument to prove this
result.
