Parametrising a circle

Problem

Parametrise the circle with centre $(1,-2)$ and radius $\sqrt 5$.

Solution

Use the standard parametrisation of a circle with centre $(x_0,y_0)$ and radius $r$: $x=x_0+r\cos t$, $y=y_0+r\sin t$. With $(x_0,y_0)=(1,-2)$ and $r=\sqrt 5$: $\begin{cases} x(t) = 1 + \sqrt 5\,\cos t \\ y(t) = -2 + \sqrt 5\,\sin t\end{cases}, \quad t\in[0,2\pi).$ Check: $(x-1)^2 + (y+2)^2 = 5(\cos^2 t + \sin^2 t) = \boxed{5}$.

Links

Topics: Complex numbers
Concepts: Circle · Parametrisation
Skills: Modelling
Exercise type: Modelling problem