Conic Sections
    Ellipse and Line
      Intersection of ellipse and line - tangency condition
      Equation of the tangent at a point on the ellipse
          Construction of the tangent at a point on the ellipse
      Tangents to an ellipse from a point outside the ellipse - use of the tangency condition
         Construction of tangents from a point outside the ellipse
      Ellipse and line examples
Intersection of ellipse and line - tangency condition
Common points of a line and an ellipse we find by solving their equations as a system of two equations in two unknowns, x and y,
                                       (1)  y = mx +                
                                       (2)  b2x2 + a2y2 = a2b2
                                                                           
                   by plugging (1) into (2)    =>   b2x2 + a2(mx + c)2a2b2
                             after rearranging,        (a2m2 + b2)x2 + 2a2mcx + a2c2 - a2b2 = 0
obtained is the quadratic equation in x. Thus, the coordinate of intersections are,
Using the above solutions follows that a line and an ellipse can have one of three possible mutual positions
depending of the value of the discriminant 
D = a2m2 + b2 - c2. Thus, if
     D > 0,  a line and an ellipse intersect,
    D = 0,  or a2m2 + b2 = c2    a line is the tangent of the ellipse and it is tangency condition.
                    The line touches the ellipse at the tangency point whose coordinates are:
     D < 0,  a line and an ellipse do not intersect.
Equation of the tangent at a point on the ellipse
In the equation of the line  y - y1 = m(x - x1) through a given point P1, the slope m can be determined using known coordinates (x1, y1) of the point of tangency, so
b2x1x + a2y1y = b2x12 + a2y12, since  b2x12 + a2y12 = a2b2  is the condition that P1 lies on the ellipse
  then b2x1x + a2y1y = a2b2 is the equation of the tangent at the point P1(x1, y1) on the ellipse.
Construction of the tangent at a point on the ellipse  
Draw a circle of a radius a concentric to the ellipse. Extend the ordinate of the given point to find intersection
with the circle. The tangent of the circle at Pc  intersects  the x-axis at Px. The tangent to the ellipse at the point     P1on the ellipse intersects the x-axis at the same point. 
  To prove this, find the x-intercept of each tangent         
analytically.                                                                   
  Therefore, in both equations of tangents set  y = 0 and 
solve for
x,
it is the x-intercept of the tangent tc and the tangent te.  
Tangents to an ellipse from a point outside the ellipse - use of the tangency condition
Coordinates of the point A(x, y), from which we draw tangents to an ellipse, must satisfy equations of the tangents,  y = mx + c and their slopes and intercepts, m and c, must satisfy the condition of tangency therefore, using the system of equations,
      (1)  y = mx + c     <=    A(x, y)
      (2)  a2m2 + b2 = c2    determined are equations of the tangents from a point A(x, y) outside the ellipse.
                                      
Construction of tangents from a point outside the ellipse
With A as center, draw an arc through F2, and from F1 as center, draw an arc of the radius 2a. Tangents are
then the perpendicular bisectors of the line segments, F2S1 and F2S2.  
  We can also draw tangents as lines through A and    the intersection points of the segments F1S1 and F1S2 and the ellipse.   
Thus, these intersections are the tangency points of     the tangents to the ellipse.  
Explanation of the construction lies at the fact that,     
F1S1 = F1D1 + D1S1 = F1D1 + D1F2 = 2a
according to the definition of the ellipse, as well as the point A is equidistant from points F2 and S1, since the point S1 lies on the arc drawn from A through F2.       
Ellipse and line examples
Example:  At a point A(-c, y > 0) where c denotes the focal distance, on the ellipse 16x2 + 25y2 = 1600 drawn is a tangent to the ellipse, find the area of the triangle that tangent forms by the coordinate axes.
Solution:   Rewrite the equation of the ellipse to the standard form  16x2 + 25y2 = 1600 |  1600
We calculate the ordinate of the point A by plugging the abscissa into equation of the ellipse                      
x = -6  =>    16x2 + 25y2 = 1600,
or, as we know that the point with the abscise             x = - c  has the ordinate equal to the value of the         semi-latus rectum,
The area of the triangle formed by the tangent and the coordinate axes,
Example:  Find a point on the ellipse x2 + 5y2 = 36 which is the closest, and which is the farthest from the line 6x + 5y - 25 = 0.
Solution:   The tangency points of tangents to the ellipse which are parallel with the given line are, the 
closest and the farthest points from the line.
Rewrite the equation of the ellipse to determine its axes,
Tangents and given line have the same slope, so
Using the tangency condition, determine the intercepts c,
therefore, the equations of tangents,
Solutions of the system of equations of tangents to the ellipse determine the points of contact, i.e., the 
closest and the farthest point of the ellipse from the given line, thus
Example: Determine equation of the ellipse which the line -3x + 10y = 25 touches at the point P(-3, 8/5).
Solution:   As the given line is the tangent to the ellipse, parameters, m and c of the line must satisfy the tangency condition, and the point P must satisfy the equations of the line and the ellipse, thus
Intermediate algebra contents
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