Conic Sections
    Ellipse and Line
      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
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: 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
Conic sections contents
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