# 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.

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