
Parametric Equations 
The parametric equations of a
circle

The parametric equations of a
circle
centered at the origin with radius
r 
The parametric equations of a
translated circle with center (x_{0}, y_{0}) and radius r 
The parametric equations of an
ellipse 
The parametric equations of an
ellipse
centered at the origin 
The parametric equations of a
translated ellipse with center at (x_{0}, y_{0}) 






The parametric equations of a
circle

The parametric equations of a
circle
centered at the origin with radius r 
The parametric equations
of a
circle
centered at the origin 
with radius
r, 

where, 0
< t < 2p. 
To
convert the above equations into Cartesian
coordinates, 
square
and add both equations, so we get 
x^{2}
+ y^{2}
= r^{2} 
as sin^{2}
t
+ cos^{2}
t = 1. 




The parametric equations of a
translated circle with center (x_{0}, y_{0}) and radius r 
The parametric equations
of a
circle with center 
(x_{0},
y_{0})
and
radius r, 
x
= x_{0}
+ r cos t 
y
= y_{0}
+ r sin t 


where, 0
< t < 2p. 
If
we write
the above equations, 
x
 x_{0}
= r cos t 
y
 y_{0}
= r sin t 
then
square and add them we get the
equation 
of the
translated circle in Cartesian coordinates, 
(x
 x_{0})^{2}
+
(y
 y_{0})^{2}
= r^{2}. 




The parametric equations of an
ellipse 
The parametric equations of an ellipse
centered at the origin 
Recall
the construction of a point of an ellipse using two concentric circles of radii equal to lengths of the 
semiaxes a and b, with the center at the origin as shows 
the
figure,
then 

where, 0
< t < 2p. 
To
convert the above parametric equations into Cartesian 
coordinates, divide the first
equation by
a
and the second 
by b, then square and add them, 

thus,
obtained is the standard equation of the ellipse. 




The parametric equations of a
translated
ellipse with center at (x_{0}, y_{0}) 
The parametric equations
of a translated ellipse with
center (x_{0},
y_{0})
and semiaxes a and b, 

x
= x_{0}
+ a cos t 
y
= y_{0}
+ b sin t 



To
convert the above parametric equations into Cartesian
coordinates, we write them as 

x
 x_{0}
= a cos t 
y
 y_{0}
= b sin t 



and divide the first
equation by
a
and the second by b, then square and add
both equations, so we get 











Functions
contents B




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