Ellipse and Line
      Angle between the focal radii at a point of 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
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.
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
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:  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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