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 xy-plane
 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 P1(x1, y1) and the direction vector s then, the position or
(radius) vector r of any point P(x, y) of the line
r = r1 + t s,    - oo < t < + oo   and where,  r1 = x1i + y1 j  and  s = xsi + ys 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 = (x1 + xst) i + (y1 + yst) 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 = x1 + xs t
y = y1 + ys 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 P1(x1, y1) and P2(x2, y2)
x = x1 + (x2 - x1) t
y = y1 + (y2 - y1) 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 x-intercept as the given point P1, so
for   y = 0   =>   0 = (-1/2)x + 3,   x = 6  therefore,  P1(6, 0).
Substitute the values, x1 = 6y1 = 0, xs = 2,  and  ys = -1 into the parametric equations of a line
x = x1 + xs t,      x = 6 + 2t
  y = y1 + ys 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 x1 = -2y1 = 0, x2 = 2,  and  y2 = 2 into the parametric equations of a line
                  x = x1 + (x2 - x1) t,      x = -2 + (2 + 2) t = -2 + 4t,       x = -2 + 4t,
                                   y = y1 + (y2 - y1) 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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