  In this paper we consider nonlocal energies defined on probability measures
in the plane, given by a convolution interaction term plus a quadratic
confinement. The interaction kernel is $-\log|z|+\alpha\, x^2/|z|^2, \;
z=x+iy,$ with $-1 < \alpha< 1.$ This kernel is anisotropic except for the
Coulombic case $\alpha=0.$ We present a short compact proof of the known
surprising fact that the unique minimiser of the energy is the normalised
characteristic function of the domain enclosed by an ellipse with horizontal
semi-axis $\sqrt{1-\alpha}$ and vertical semi-axis $\sqrt{1+\alpha}.$ Letting
$\alpha \to 1^-$ we find that the semicircle law on the vertical axis is the
unique minimiser of the corresponding energy, a result related to interacting
dislocations, and previously obtained by some of the authors. We devote the
first sections of this paper to presenting some well-known background material
in the simplest way possible, so that readers unfamiliar with the subject find
the proofs accessible
