Note on a symmetric Diophantine equation

Using an elementary argument, we show that for all rational numbers α such that neither α nor 3α is a rational square, the equation x 4 -4αx 2 -4αy 2 +y 4 =-6α 2 has no rational solutions. This answers Hindes’ two questions and generalizes his theorem (Theorem 1.1) in “Rational points on certain families of symmetric equations”, Int. J. Number Theory 11 (2015), no. 6, pp. 1821–1838.