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.

Hello,

f(x,y)=x²-xy+y²-3=0

If y=0 , x²-3=0 ==>x=-√3 or x=√3

Let's calculate dy/dx
@ means derivate partial.

@f/@x=2x-y
@f/@y=-x+2y

dy/dx=-(@f/@x) /(@f/@y)=-(2x-y)/(-x+5y)

if y=0 and x=√3 then dy/dx=-2√3 / (-√3)=2
idem if y=0 and x=-√3

Equations of the tangent lines

In (√3,0), y-0=2(x-√3)
In (-√3,0), y-0=2(x+√3)