x2 y2 -- - --- = 1 a2 b2In this equation a and b are strictly positive real numbers. The equation consists of two functions.
x2 y2 -- - --- = 1 a2 b2 <=> b2 (x2 - a2 ) y2 = ----------------- a2 <=> _________ _________ b | 2 2 b | 2 2 y = - \| x - a or y = - - \| x - a a aThe graph of the first one is above the x-as and the second graph is the reflection of the first graph in the x-axis. If we calculate the asymptotes a and a', we find
b b y = - x and y = - - x a a
The intersection points of the hyperbola with the x-axis are A'(-a,0) and
A(a,0). These are the vertices on the x-axis.
The segment [A',A] is called major axis of the ellipse.
The segments [P,F'] and [P,F] are the focal radii through point P.
x2 - y2 = a2This hyperbola is called an orthogonal hyperbola.
x = a sec(t) (1) y = b tan(t) (2)The real number t is the parameter.
a x = ------ cos(t) b sin(t) y = ------- cos(t)or to
a cos(t) = - x a y sin(t) = --- b xThis system has a solution for t if and only if
sin2 (t) + cos2 (t) = 1 <=> a 2 a y 2 (-) + (---) = 1 x b x <=> a2 b2 + a2 y2 = b2 x2 <=> x2 y2 -- - --- = 1 a2 b2Hence, the two associated lines constitute a curve and that curve is the hyperbola.
P(a sec(t) , b tan(t))is on the hyperbola for each t-value and with each point of the hyperbola corresponds a t-value.
|PF|2 = (x-c)2 + y2 a b2 sin2t = ( ----- - c )2 + ----------- cos t cos2t 1 = --------- ( a2 - 2 a c cos t + c2 cos2 t + (c2-a2) sin2t ) cos2 t 1 = --------- ( a2 cos2t - 2 a c cos t + c2) cos2 t 1 = --------- ( a cos t - c)2 cos2 t |PF|2 = (x+c)2 + y2 = .... 1 = --------- ( a cos t + c)2 cos2 t Since c > a we have |PF| = (1/cos(t)).(c - a cos(t)) |PF'| = (1/cos(t)).(c + a cos(t)) |PF'| - |PF| = 2aIn the same way as for the ellipse you can show that if the difference of the distances from P to F and F' is equal to 2a, the point P is on the hyperbola. (exercise)
|The hyperbola is the locus of the point P such that the difference of the distances from P to F and F' is equal to 2a|
xo x yo y ---- - ---- = 1 a2 b2It is the bisector t of the lines PF and PF'.
A hyperbola H has vertices P and P' and D is on H.|
Prove that the product of the slopes of DP and DP' is constant.
Calculate the lines with slope = 1 and tangent to the hyperbola
9 x2 - 25 y2 = 225Find the equation of the line connecting the points of tangency.
A point P(xo,yo) is on the hyperbola H with foci F and F'.|
Prove that |D,F| = | a - (c/a) xo |.
|Find the equations of the tangent lines, with slope m, to a hyperbola.|
|Point P is on the hyperbola x2/a2 - y2/b2 = 1. Take on an asymptote s point Q with the same abscissa as point P. The line l is the perpendicular on s through point Q and the line n is the normal line in point P. Show that l and n intersection on the x-axis.|
Find the m-values such that the angle between the asymptotes of the hyperbola|
x2/(m+1)2 - y2/m2 = 1
is equal to 60 degrees.
Point P is on the orthogonal hyperbola x2 - y2 = a2.|
Point P' is the perpendicular projection of P on the x-axis.
Show that |PP'|2 is equal to the power of point P' relative to a circle x2+y2=a2.
Prove that the slopes m of the lines through P(xo,yo) and tangent
to the hyperbola
b2 x2 - a2 y2 = a2 b2 are the solutions of the equation (xo2 - a2) m2 - 2 xo yo m + yo2 + b2 = 0
The tangent line in point P of a hyperbola has with the
asymptotes the intersection points Q and Q'.|
Show that P is the midpoint of the segment [Q,Q'].