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
      Angle between the focal radii at a point of 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.  
Angle between the focal radii at a point of the ellipse
Let prove that the tangent at a point P1 of the ellipse is perpendicular to the bisector of the angle between the focal radii r1 and  r2.
Coordinates of points, F1(-c, 0), F2(c, 0) and P1(x1, y1) plugged into the equation of the line through two given points determine the lines of the focal radii 
r1 = F1P1  and   r2 = F2P1,
and the equation of the tangent at the point P1,
By plugging the slopes of these tree lines into the formula for calculating the angle between lines we find the
exterior angles
j1 and j2 subtended by these lines at P1.
Thus, using the condition b2x12 + a2y12 = a2b2, that the point lies on the ellipse, obtained is
If on the same way we calculate the interior angle subtended by the focal radii at P1, and which is the supplementary angle of the angle j,
then compare with the result which will we obtain by using the double-angle formula for the angles j1 and j2,
To compare obtained results, we multiply both the numerator and the denominator of the result for the supplementary angle by b2,
what proved the previous statement.
Therefore, the normal at the point P1 of the ellipse bisects the interior angle between its focal radii.
Ellipse and line examples
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
Example:  At which points curves, x2 + y2 = 8 and  x2 + 8y2 = 36, intersect? Find the angle between two curves.
Solution:   Given curves are the circle and the ellipse. The solutions of the system of their equations determine the intersection points, so
Angle between two curves is the angle between      tangents drawn to the curves at their point of            intersection.
The tangent to the circle at the intersection S1,
S1(2, 2)  =>   x1x + y1y = r2,   2x + 2y = 8
or  tc ::   y = - x + 4    therefore,  mc = -1.
The tangent to the ellipse at the intersection S1,
The angle between the circle and the ellipse,
Example:  The line x + 14y - 25 = 0 is the polar of the ellipse x2 + 4y2 = 25. Find coordinates of the pole.
Solution:   Intersections of the polar and the ellipse are points of contact of tangents drawn from the pole P to ellipse, thus solutions of the system of equations,
(1)  x + 14y - 25 = 0   =>   x = 25 - 14y  =>  (2)
    (2)  x2 + 4y2 = 25
                                    
   (25 - 14y)2 + 4y2 = 25
2y2 - 7y + 6 = 0,      y1 = 3/2 and  y2 = 2
y1 and y2  =>  x = 25 - 14y,   x1 = 4 and  x2 = -3.
Thus, the points of contact  D1(4, 3/2) and D2(-3, 2).
The equations of the tangents at D1 and D2,
The solutions of the system of equations t1 and t2 are the coordinates of the pole P(1, 7/2).
Example:  Find the equations of the common tangents of the curves  4x2 + 9y2 = 36 and  x2 + y2 = 5.
Solution:  The common tangents of the ellipse and the circle must satisfy the tangency conditions of these curves, thus
Pre-calculus contents H
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