Parametric Equations

We can define a curve  
  para metrically by introducing a convenient parameter which defines a point on a curve. The parameter may define for example the length along the curve from a given point, and this is useful for some purposes.
Example: We can define a circle radius  
  centred at the point  
  in terns of the parameter  
, the angle between the horizontal and a radius from the centre of the circle at  
  to a point  
  on the circle  
\[x=2+5 cos \theta , \; y=3+5 sin \theta\]
. In this example the parameter  
  is not the length from a given point.
We can write the line  
  para metrically as  
\[(\frac{t}{ \sqrt{5}} , \frac{2}{ \sqrt{2}} +3 )\]
. In this example as  
  increases by 1, we move  
\[\sqrt{(\frac{1}{\sqrt{5}})^2+ ( \frac{2}{\sqrt{5}})^2}=1\]
  which is the distance along the curve.
Parametric equations are useful for more complex curves, where it could be hard to give the equation in the form  
\[(x,y)=(t^2e^{-t}-t, 2+\frac{1+e^t}{2-t} )\]

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