Coordinate geometry or Analytic geometry
     Line in a coordinate plane
      The parametric equations of a line
      Parallel and perpendicular lines
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's 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's 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 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,  
Parallel and perpendicular lines
Two lines having slopes m1 and m2 are parallel if 
  m1 = m2   that is, if they have the same slope.  
To find the criteria two lines,  y = m1and  y = m2x to be perpendicular or orthogonal we can use the principle of similar triangles, OA'A and OB'B shown in the picture.
Therefore, m1 : 1 = -1 : m2 =>
This relation will stay unchanged if we translate the perpendicular lines, that is, when lines 
              y = m1x + c1 and  y = m2x + care written in the slope-intercept form.
Two lines are perpendicular if the slope of one line is the negative reciprocal of the other.
Example:  Find the equation of the line that is perpendicular to the line   and passes through 
the point A(-2, 5).
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