Questions about ellipse

Question

Given the center of an ellipse and three of its points, is this ellipse completely determined?

What is the easiest way to show that five points of an ellipse are enough to determine the ellipse?

Answer 1

Hoping that I am not wrong : the general equation of the ellipse write $$Ay^2+B x y+C x^2+D y + E x+ F=0$$ Dividing by $A$, it then reduces to $$y^2+\alpha xy+\beta x^2+\gamma y +\delta x+\epsilon=0$$ Since we know the center $(x_0,y_0)$, make a change of variable $x=x_0+X$, $y=y_0+Y$ (in order to eliminate the $x$ and $y$ terms) and it becomes $$Y^2+\alpha' XY+\beta'X^2+\gamma'=0$$ So, three equations are at least required, so three points if you know the center.

Answer 2

The answer to the first part is: NO. Let us consider 4 edges of a square. There are infinitely many ellipses going through these 4 points.