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# Circle and ellipse

The distance $d$ between two points $(x,y)$ and $(x_0,y_0)$ is $$d\big((x,y),(x_0,y_0)\big) = \sqrt{(x-x_0)^2+(y-y_0)^2}.$$

The equation of a circle centred at $(x_0,y_0)$ with radius $r\gt 0$ is $$(x-x_0)^2+(y-y_0)^2 = r^2.$$ It consists of all the points that have a distance $r$ to the centre, since the above identity can be written as $d\big((x,y),(x_0,x_0)\big) = r$.

The equation of an ellipse centred at $(x_0,y_0)$ with radius $a$ and $b$ is $$\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2} = 1$$

The ellipse passes through the points $(x_0\pm a,y_0)$ and $(x_0,y_0\pm b)$.

An ellipse is the result of dilating a circle of radius 1 by a ratio of $a$ along the $x$-axis and a ratio of $b$ along the $y$-axis.

Circle (A) and ellipse (B)