Coordinate geometry or Analytic geometry
     Line in a coordinate plane
      Hessian normal form of the equation of a line
         Distance between a point and a line
      Lines and points relations, examples
Hessian normal form of the equation of a line
The normal p from the origin to a line subtends the angle j with the x-axis and form two right triangles with axes as is shown in the figure below, then
Hessian normal form of the equation of a line.
By comparing the normal form with the general form Ax + By + C = 0, of the equation of the line, we can find relations between corresponding coefficients of both forms so, if we write
lAx + lBy + lC = 0
   and after comparing both equations,      lA = cosj   and  lB = sinj
then, by plugging into the general form obtained is
the normal form of the equation of a line expressed by coefficients of the general form, where the sign of the square root should be opposite to the sign of C.
Distance between a point and a line
The distance from a point A(x0, y0) to a line l, equals the distance between the given line and a line passing through the given point parallel to the given line, as shows the figure below.
So, the length of the normal from the origin to that new line is  p + d, and the coordinates of A will then satisfy equation of the line, that is
x0 cos j + y0 sin j p + d    or
d = x0 cos j + y0 sin j - p.
If equation of a line is given in the general form, then the distance is
The sign of the square root is taken to be opposite to sign of C, and where the negative result shows that the given point is at the same side as the origin in relation to the given line.
Example:  Given is a point A(-2, 3) and the direction vector of a line s = i - 2j, determine equation of the line through the point A and draw a diagram.
Solution:  Plug the point A and the components of the direction vector into the parametric equation of a line:
Example:  Write equation of a line in the point slope form and in parametric form if the line passes through the point A(2, -1) and whose slope  m = -3, draw a diagram.
Solution:  Plug the given elements of the line into equation
Example:  A line passes through points A(-3, -4) and B(9, 4). Find the general form of its equation and the segments that the line cuts on the coordinate axes.
Solution:  From the equation of a line through two points
Example:  Check if points, A(2, -1), B(5, -5) and C(-1, 3) all lie on the same line.
Solution:  If given points lie on the same line then must be satisfied following condition
Check can also be done by examining if given points satisfy vector’s equation
by taking coordinates of any of three given points as the coordinates of the radius vector r.
Example:  A line through a point A(-3, 5) forms a triangle with the coordinate axes whose area is 24 square units, find equation of the line.
Solution:  Area of a triangle    
Coordinates of the point A must satisfy equation of the line 
The solution  n = 4 satisfies given conditions, so that 
Example:  Find the distance of the line  -4x + 3y - 25 = 0  from the origin.
Solution:  Rewrite the equation of given line from the general form to the normal form, that is
Comparing with the normal form  xcosj + ysinj - p = 0,  follows that the distance  p = 5.
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