I started thinking about it, but work got in the way. Checking back a little later I saw an amazing solution (that fits in a tweet!) from Dave Radcliffe:

@jamestanton Yes! Solve the equations for x^2 and y^2 respectively, then add the equations to obtain the equation of a circle. Nice problem!

You might enjoy tinkering around with this slight generalization. Linear combination rocks! https://www.desmos.com/calculator/vkvaesiz8s