Home page for accesible maths 3 Chapter 3 contents 3.3 Tangent vector to a curve 3.5 Length of a curve

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3.4 Parameters for circles

Example.

Let $C$ be the circle with diameter one and centre $(1/2,0)$ , so that the $y$ –axis is tangent to $C$ . Find a parametric form for $C$ and the rate of change of $(x(t),y(t))$ with respect to $t$ .

Solution. Since the circle has diameter $1$ , it has radius $\frac{1}{2}$ . As the centre is $(\frac{1}{2},0)$ , we can apply the parametrisation of the unit circle to the coordinates $(2(x-\frac{1}{2}),2y)$ . Using the statement in frame 3.2, we have:

x=\,{\frac{1}{2}+\frac{1}{2}\frac{1-t^{2}}{1+t^{2}}}\,{=\frac{1}{2}\frac{1+t^{% 2}+1-t^{2}}{1+t^{2}}}\,{=\frac{1}{1+t^{2}}}

\mbox{and}\;\;y=\,{\frac{1}{2}\frac{2t}{1+t^{2}}}\,{=\frac{t}{1+t^{2}}.}