  The collective excitation modes in quantum droplets trapped in a
two-dimensional harmonic potential in the context of symmetric weakly
interacting binary bosonic mixtures are studied. By utilizing the linearization
technique, the time-dependent extended Gross-Pitaevskii equation, and a
sum-rule approach with a variational approximation, the ground state properties
and collective excitations of such a two-dimensional quantum system are
investigated for various system parameters. We present comprehensive analysis
and calculations on the effect of the confinement strength and anisotropy of
the trapping potential, the number of atoms in the droplet, and the collective
excitation modes. The radius of the droplet, as well as the chemical potential,
is non-monotonically related to the number of atoms in the droplet, and the
confinement tends to shift the minimum values towards the ideal gas limit. The
excitation frequency peaks, which are prominent in a self-bounded droplet,
become less pronounced and smoother when subjected to a strong trapping
potential. The sum-rule approach fails to reproduce the breathing mode
frequency for a moderate number of atoms in a weak trapping potential, however,
works perfectly well in a strong confinement. It was found that the anisotropy
in the trap eliminates the degeneracy between the quadrupole and scissors modes
that occurs in an isotropic trap, causing the frequencies of these two modes to
immediately diverge from each other for any degree of anisotropy. These
findings provide valuable insights into the unique characteristics and behavior
of quantum droplets, offering potential implications for future research and
applications in the dynamic behaviors of intriguing quantum droplets.
