Conic Sections
     Hyperbola and Line
      Hyperbola and line relationships
         Condition for a line to be the tangent to the hyperbola - tangency condition
      The equation of the tangent at the point on the hyperbola
      Polar and pole of the hyperbola
      Construction of the tangent at the point on the hyperbola
      Construction of tangents from a point outside the hyperbola
      Hyperbola and line examples
Hyperbola and line relationships
Let examine relationships between a hyperbola and a line passing through the center of the hyperbola, i.e., the origin. A line y = mx intersects the hyperbola at two points if
     the slope   |m| < b/a  but if  |m| > b/a  then,       
the line y = mx does not intersect the hyperbola at all.
The diameters of a hyperbola are straight lines passing through its center.
The asymptotes divide these two pencils of diameters into, one which intersects the curve at two points, and  the other which do not intersect.
A diameter of a conic section is a line which passes    
through the midpoints of parallel chords.
Conjugate diameters of the hyperbola (or the ellipse)    
are two diameters such that each bisects all chords     drawn parallel to the other.                                         
As the equation of a hyperbola can be obtained from the equation of an ellipse by changing the sign of b2  
that is,     a2(1 - e2) = -a2(e2 - 1) = -b2
this way, we can use other formulas relating to the ellipse to obtain corresponding formulas for the hyperbola.
Therefore, when we examine conditions which determine position of a line in relation to a hyperbola that is, 
when solve the system of equations,             y = mx + c
b2x2 - a2y2 = a2b2
                                                                                              
           then if,    a2m2 - b2 > c2   the line intersects the hyperbola at two points,
                         a2m2 - b2 = c2   the line is the tangent of the hyperbola,
                         a2m2 - b2 < c2   the line and the hyperbola do not intersect.
Condition for a line to be the tangent to the hyperbola - tangency condition
A line is the tangent to the hyperbola if a2m2 - b2 = c2.  
Regarding the asymptotes, to which c = 0, this condition gives |m| = b/a and that is why we can say that the hyperbola touches the asymptotes at infinity.
From the tangency condition it also follows that the slopes of the tangents will satisfy the condition if
That is, the tangents to the hyperbola can only be parallel to a line belonging to the pencil of lines that do not intersect the hyperbola.
The equation of the tangent at the point on the hyperbola
As we already mentioned, the points of contact of a line and the hyperbola can be obtained from the 
corresponding formula for the ellipse by changing
b2 with -b2 thus
  the tangency point or the point of contact.
So, the intercept and slope of the tangent
or b2x1x - a2y1y = a2b2  the equation of the tangent at a point P1(x1, y1) on the hyperbola.
Polar and pole of the hyperbola
If from a point A(x0, y0), exterior to the hyperbola, drawn are tangents, then the secant line passing through 
the contact points, is the polar of the point
A. The point A is called the pole of the polar.
The equation of the polar is derived the same way as for the ellipse,
  b2x0x - a2y0y = a2b2  the equation of the polar of the point A(x0, y0).
Construction of the tangent at the point on the hyperbola
The tangent at the point P1(x1, y1) on the hyperbola is the bisector of the angle F1P1F2 subtended by focal 
radii, r1 and r2 at P1 .
The proof shown for the ellipse applied to the          hyperbola gives,                                          
or  
See the title ' The angle between the focal radii at a point of the ellipse'.
Construction of tangents from a point outside the hyperbola
With A as center draw an arc through F2, and from F1as center, draw an arc of radius 2a.  
These arcs intersect at points S1 and S2.               
Tangents are the perpendicular bisectors of the line segments F2S1 and F2S2.                                   
Tangents can also be drawn as lines through A and the intersection points of lines through F1S1 and     F1S2, with the hyperbola.                                     
These intersections are at the same time the points of contact D1 and D2.                                          
Hyperbola and line examples
Example:  Determine the semi-axis a such that the line 5x - 4y - 16 = 0 be the tangent of the hyperbola
9x2 - a2y2 = 9a2.
Solution:   Rewrite the equation          9x2 - a2y2 = 9a2 | 9a2
and the equation of the tangent  5x - 4y - 16 = 0  or
Then, plug the slope and the intercept into tangency condition,
Therefore, the given line is the tangent of the hyperbola
Example:  Find the angle between the ellipse, which passes through points,  A(5, 4/3) and B(1, 42/3), and the hyperbola whose asymptotes are y = x/ 2 and the linear eccentricity or half of the focal distance 
c = 5.
Solution:  Find the equation of the ellipse by solving the system of equations,
thus, the equation of the ellipse
The equation of the hyperbola by solving the system of equations,
Therefore, equation of the hyperbola
Angle between curves is the angle between tangents at intersection of the curves. By solving the system of equations of the curves we obtain the points of intersection,
   (1)  4x2 + 9y2 = 36    (2)  => (1)   4(4y2 + 4) + 9y2 = 36,   25y2 = 20,   y1,2 = 2/5,   x1,2 = 6/5
   (2)    x2 - 4y2 = 4   =>   x2 = 4y2 + 4
Tangent of the ellipse and the hyperbola at the intersection S1(6/5, 2/5),
    S1(6/5, 2/5)    =>   te ::   b2x1x + a2y1y = a2b2,    4x + 3 = 6/5    or    y = (-4/3)x + 25,
  S1(6/5, 2/5)   =>   th ::   b2x1x - a2y1y = a2b2,    3x - 4 = 2/5    or    y = (3/4)x - 5/2.
fulfilled is the perpendicularity condition. Therefore, the angle between curves j = 90.
Conic sections contents
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