Conic Sections
    Parabola and Line
      Common points of a line and a parabola
      Condition for a line to be the tangent to the parabola - tangency condition
      The equation of the tangent and the normal at the point on the parabola
      Properties of the parabola
Common points of a line and a parabola
Common points of a line and a parabola we determine by solving their equations as the system of two 
equations in two unknowns,
                                                    (1y = mx + c
                    (2y2 = 2px            (1)  =>   (2)      m2x2 + 2(mc - p)x + c2 = 0,
therefore, the coordinates of intersections of a line and a parabola
Condition for a line to be the tangent to the parabola - tangency condition
In the formulas for calculating coordinates of intersections there is the expression under the square root whose value determines three possible cases regarding mutual position of a line and a parabola,
so for p > 0 and,
  p - 2mc > 0  - the line intersects the parabola at two points S1(x1, y1) and S2(x2, y2),
  p - 2mc = 0  - the line is the tangent of the parabola and have one point of contact D((p - mc)/m2, p/m))
                         or by substituting  p = 2mc, the tangency point D(c/m, 2c),
  p - 2mc < 0  - the line and the parabola do not intersect.
If we write the above conditions as then 
these three cases can be explained graphically as      the relation between parameters m and c of the line    and the position of the focus F(p/2, 0).                    
At the y-intercept N of the line drawn is a                   perpendicular which intersects the x-axis at M, then  
ON = | c | and OM = | c tan a | = | m c |.
Therefore, when the point M is located;                    
- to the left of F, i.e.,  if  m c < p/2
the line intersects the parabola at S1and S2,             
  - to the right of F, i.e.,  if  m c > p/2   
the line and the parabola do not intersect,                 
            - at the focus F or if   m c = p/2                
the line is the tangent of the parabola.                       
The equation of the tangent and the normal at the point on the parabola
In the equation of the line y - y1 = m( x - x1) through the given point we express the slope m by the given 
ordinate of the tangency point,
and since the coordinate of the tangency point must        satisfy the equation of the parabola, then                       
 obtained is   y1y = p(x + x1) the equation of  
the tangent at the point P(x1, y1) on the parabola.          
Since the above equation 
can be written using coordinates of the tangency point    
   
As the slope of the normal then the equation of the normal at P(x1, y1),
  or  
Properties of the parabola
  Using equations of the tangent and normal expressed by coordinates of the tangency point and the figure above;
   a)  y-intercept ct of the tangent equals half of the ordinates of the tangency point, ct = y1/2.
   b) the projection AB of the segment BP1 of the tangent to the x-axis, i.e., to the axis of the parabola, is 
       equal to twice the abscissa of the tangency point, so
                         AB = St = 2x- the line segment AB is called the subtangent.
   c) the projection AC of the segment CP1 of the normal to the x-axis is equal to the parameter p, i.e.,
                         AC = Sn =   - the line segment AC is called the subnormal.
As points B and C are x-intercepts of the tangent and the normal their abscissas we determine by solving 
corresponding equations for
y = 0, so
put y = 0 into equation of the tangent,
put y = 0 into equation of the normal,
Thus, the focus F(p/2, 0) bisects the line segment BC whose endpoints are x-intercepts of the tangent and the normal, as shows the figure above.
  The tangent at any point on the parabola bisects the angle j between focal distance and the perpendicular to the directrix and is equally inclined to the focal distance and the axis of the parabola. 

  The normal at the tangency point bisect the supplementary angle of the angle j.

Since,   DP1 = FP1 = r = x1 + p/2,
and       BF = FC = x1 + p/2 = r, and  DP1 || BC   
then, following triangles are congruent,                      
DBFD @ DFCP1 @ DFP1D
so, the quadrangle BFP1D is the rhombus and its
diagonal
BP1 bisects the angle j.                           
  This property is known as the reflective property of the parabola.
 A light rays coming in parallel to the axis of a parabolic mirror (telescope), are reflected so that they all pass through the focus. Similarly, rays originating at the focus (headlight) will be reflected parallel to the axis.
  Tangents drawn at the endpoints of a focal chord intersect at right angles on the directrix.
a)  Solving the system of equations of tangents, 
x is the abscissa of the intersection S of tangents.
The slope of the focal chord line through tangency points,
therefore,  x = - p/2 is the abscissa of the intersection S(- p/2, y) and the equation of the directrix d.
b)  The tangent to the parabola which passes through intersection S, which lies on the directrix, must satisfy tangency condition,
Thus, satisfied is condition for perpendicularity,   m1 m2 = - 1    =>    j = 90.
Conic sections contents
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