Question

The equation (x^2) - xy + (y^2) = 3 represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.

Answer 1

when it crosses the x-axis, y=0

Answer 2

Do a coordinate rotation to a form without the xy-term, and show that the slopes of those two points on the ellipse are equal. A rotation back into the equation rotates both lines/slopes, thus proving that the tangent lines are parallel.