x^{2} y^{2} -- - --- = 1 a^{2} b^{2}In this equation a and b are strictly positive real numbers. The equation consists of two functions.
x^{2} y^{2} -- - --- = 1 a^{2} b^{2} <=> b^{2} (x^{2} - a^{2} ) y^{2} = ----------------- a^{2} <=> _________ _________ 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.
x^{2} - y^{2} = a^{2}This 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
sin^{2} (t) + cos^{2} (t) = 1 <=> a 2 a y 2 (-) + (---) = 1 x b x <=> a^{2} b^{2} + a^{2} y^{2} = b^{2} x^{2} <=> x^{2} y^{2} -- - --- = 1 a^{2} b^{2}Hence, 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} + y^{2} a b^{2} sin^{2}t = ( ----- - c )^{2} + ----------- cos t cos^{2}t 1 = --------- ( a^{2} - 2 a c cos t + c^{2} cos^{2} t + (c^{2}-a^{2}) sin^{2}t ) cos^{2} t 1 = --------- ( a^{2} cos^{2}t - 2 a c cos t + c^{2}) cos^{2} t 1 = --------- ( a cos t - c)^{2} cos^{2} t |PF|^{2} = (x+c)^{2} + y^{2} = .... 1 = --------- ( a cos t + c)^{2} cos^{2} 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 a^{2} b^{2}It 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 x^{2} - 25 y^{2} = 225Find the equation of the line connecting the points of tangency. |
A point P(x_{o},y_{o}) is on the hyperbola H with foci F and F'. Prove that |D,F| = | a - (c/a) x_{o} |. |
Find the equations of the tangent lines, with slope m, to a hyperbola. |
Point P is on the hyperbola x^{2}/a^{2} - y^{2}/b^{2} = 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 x^{2}/(m+1)^{2} - y^{2}/m^{2} = 1 is equal to 60 degrees. |
Point P is on the orthogonal hyperbola x^{2} - y^{2} = a^{2}. 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 x^{2}+y^{2}=a^{2}. |
Prove that the slopes m of the lines through P(x_{o},y_{o}) and tangent
to the hyperbola
b^{2} x^{2} - a^{2} y^{2} = a^{2} b^{2} are the solutions of the equation (x_{o}^{2} - a^{2}) m^{2} - 2 x_{o} y_{o} m + y_{o}^{2} + b^{2} = 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']. |