Definition: An ellipse is a curve on a plane surrounding two focal points such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve.
I can see why $\frac{x^2}{a^2}+\frac{y^2}{b^2} =1 $ is an ellipse.
But why does the equation $x^2 + xy +y^2 =1 $ represents an ellipse ? Can someone please show me the calculations involved?
Thanks in advance!
Hint: Let $u=x-y$ and $v=x+y.$ Note that $u,v$ are orthogonal (have a dot product of $0$), just as $x,y$ are, and then try to rewrite the equation in the form $$\frac{u^2}{a^2}+\frac{v^2}{b^2}=1$$ for some appropriately chosen $a,b.$ Ultimately, this is an ellipse of the usual sort that has been rotated by $45^\circ.$
Standard equation of a 2 degree curve is:
$ax^2 + by^2 + 2hxy + +2gx + 2fy + c = 0$
You have to calculate $\Delta = abc + 2fgh + af^2 + bg^2 + ch^2$
If $\Delta \neq 0$ and $h^2 < ab$, then it is ellipse.
In this case, $a= 1, h = \frac12, b = 1, c = -1, g = f = 0$
$\Delta = -\frac54 \neq 0$
Also $h^2 < ab$
Thus it is an ellipse
