Accuracy of Implicit Curve Rendering
I noticed that plotting the hyperbola eq1: x² + 2 / sqrt(3) x y - y² = 1 shows the x-intercepts at about -1.00000115 and 1.00000115 instead of exactly -1 and 1. Using the Intersect tool to show the intercept points Intersect(eq1, xAxis) correctly shows them at (-1, 0) and (1, 0), so that calculation is happening correctly behind the scenes, but something is going amiss with the display of the curve itself.
Plotting the same hyperbola as Curve((sqrt(1 / 2 sqrt(3) sec(2 (θ - π / 12))); θ), θ, 0, 2π) also works correctly, visually showing the x-intercepts at exactly -1 and 1, so the problem seems to be specifically with the implicit curve rendering rather than with the hyperbola itself.
I tested this in the web application and in the desktop program. Both showed the same problem.