Conic Sections
    Parabola and Line
      The equation of the tangent and the normal at the point on the parabola
         Properties of the parabola
      Polar 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),
Properties of the parabola
  Using equations of the tangent and normal expressed by coordinates of the tangency points 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,                      
so, the quadrangle BFP1D is the rhombus and its
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,.e.,
Thus, satisfied is condition for perpendicularity,   m1 m2 = - 1    =>    j = 90.
Polar of the parabola
The polar p of a point A(x0, y0), exterior to the parabola y2 = 2px, is the secant through the contact points of the tangents drawn from the point A to the parabola.
The tangency points D1(x1, y1) and D2(x2, y2)  and the point A satisfy the equations of tangents,                      
t1 ::   y1y0 = p(x0 + x1)  and  t2 ::  y2y0 = p(x0 + x2).
Subtracting t2 - t1,
y0(y2  - y1) = p[(x0 + x2) - (x0 + x1)]
obtained is the slope of the polar. By plugging the slope into equation of the line through the given point               
or   y0y = y1y0 + px - px1
since  y1y0 = p(x0 + x1),
then y0y = p(x + x0) the equation of the polar.  
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