

Parametric Equations 
Parametric
equations definition 
The parametric equations of a line

The
parametric equations of a line passing through two points 
The direction of
motion of a parametric curve 
Evaluation
of parametric equations for given values of the parameter 
Sketching parametric
curve 
Eliminating the
parameter from parametric equations 
Parametric
and rectangular forms of equations
conversions 





Parametric
equations definition 
When
Cartesian coordinates of a curve or a surface are represented as
functions of the same variable (usually written t),
they are
called the parametric equations. 

Thus,
parametric equations in the xyplane 
x
= x (t)
and y
= y (t) 
denote
the x
and y
coordinate of the graph of a curve in the plane. 

The parametric equations of a line

If
in a coordinate plane a line is defined by the point P_{1}(x_{1},
y_{1}) and the
direction vector s
then, the position or 
(radius)
vector r
of any point P(x,
y) of the line 
r
= r_{1} + t · s,

oo
< t < + oo
and
where, r_{1}
= x_{1}i + y_{1} j
and s
= x_{s}i + y_{s} j, 
represents the vector equation of the line. 
Therefore,
any point of the line can be reached by the 
radius vector 
r
= xi + y j
= (x_{1} + x_{s}t) i
+ (y_{1} + y_{s}t) j 
since the scalar quantity t
(called the
parameter) can take 
any real value from 
oo
to + oo. 
By
writing the scalar components of the above vector 
equation obtained is 
x
= x_{1} + x_{s} · t 
y
= y_{1} + y_{s} · t 


the parametric
equations of the line. 



To
convert the parametric equations into the Cartesian coordinates solve
given equations for t.
So 

by
equating 


Therefore,
the parametric equations of a line passing through
two points P_{1}(x_{1},
y_{1}) and P_{2}(x_{2},
y_{2}) 
x
= x_{1} + (x_{2} 
x_{1}) t 
y
= y_{1} + (y_{2} 
y_{1}) t 



Parametric
curves have a direction of motion 
When
plotting
the points of a parametric curve by increasing t, the
graph of the function is traced
out in the 
direction of motion. 

Example: Write
the parametric equations of the line y
= (1/2)x
+ 3 and sketch its graph. 
Solution: Since 



Let
take the xintercept as
the given point P_{1}, so 
for
y
= 0 => 0 = (1/2)x
+ 3, x
= 6 therefore,
P_{1}(6,
0). 
Substitute
the values, x_{1}
= 6, y_{1}
= 0, x_{s}
= 2,
and y_{s}
= 1
into the parametric equations of a line 
x
= x_{1} + x_{s} · t,
x
= 6 + 2t 
y
= y_{1} + y_{s} · t,
y = t 

The direction of motion (denoted by red arrows) is given by increasing t. 

Example: Write
the parametric equations of the line through points, A(2,
0) and B(2,
2) and sketch the graph. 
Solution:
Plug the coordinates x_{1}
= 2,
y_{1}
= 0, x_{2}
= 2, and y_{2}
= 2
into the parametric equations of a line 
x
= x_{1} + (x_{2} 
x_{1}) t,
x
= 2
+ (2 + 2) t
= 2 +
4t,
x
= 2 +
4t, 
y
= y_{1} + (y_{2} 
y_{1}) t,
y = 0
+ (2 
0) t
= 2t,
y
= 2t. 
To
convert the parametric equations into the Cartesian coordinates solve 
x
= 2 +
4t for t
and plug into y
= 2t 
therefore, 












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