Hyperbolas

1) Open,planar curve that can continue upto infinity.
2) Consists of 2 disconnected curves called branches or arms.
3)The points on the two branches that are closest to each other are called Vertices. The distance of either vertice from origin is a
4) The line segment joining the vertices is called major axis or transverse axis.
5)The mid-point of the line joining the vertices iscalled the center of the hyperbola.
6) The line which is the perpendicular bisector of transverse axis passes through origin and is called conjugate axis.
7) The hyperbola has mirror symmetry about its principal axis .It is also symmetric on 180° rotation about its center.
8) At very large distances from the center,the hyperbola approaches 2 lines called asymptotes but it never intersects them.
9)A degenerate hyperbola only consists of its asymptotes.
10)If transverse axis is aligned with the x-axis of the Cartesian co-ordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign.
11)b is the length of the perpendicular from either vortex to the asymptotes.
12) A conjugate axis of length 2b, corresponding to minor axis of an ellipse is drawn on non-transverse principal axis ,its end point being +b,-b and the same lie in conjugate minor axis.
13)θ is the angle between the transverse axis and either asymptotes.
14)If b=a,2θ=90 then the hyperbola is called rectangular or equilateral hyperbola as 4 sides joined present a square.
15)Equation of a hyperbola with transverse axis along x-axis and center of hyperbola as origin is

x2/a2 - y2/b2 =1

Equation of a hyperbola with transverse axis along y-axis and center of hyperbola as origin is

y2/a2 - x2/b2 =1

16) Eccentricity of a hyperbola defined by ε is always greater than 1. ε > 1 . For ellipse, ε=1 and for circle ε<1
17) There are two foci on the transverse axis equidistant from origin.Distance of each focus from origin is c. Shape of a hyperbola is determined by its eccentricity . Eccentricity is defined by ε =c / a and c=√(a2+b2) =secθ
18) 2 lines parallel to the conjugate axis and each at a distance a/e from the origin are called Directrix.
19) Each hyperbola has a conjugate hyperbola in which transverse and conjugate axis are interchanged without changing origin and asymptotes. The equations are as in point 15)
20) If there is an angle θ such that x=acoshθ and y=bsinhθ, then  x2/a2 - y2/b2 =1  as cosh2θ-sinh2θ =1 for all - ∞ ≤θ≤. So we get all points (x,y) on the right branch of hyperbola. If there is an angle θ such that x=-acoshθ and y=bsinhθ, then  x2/a2 - y2/b2 =1  as cosh2θ-sinh2θ =1 for all - ∞ ≤θ≤. So we get all points (x,y) on the left  branch of hyperbola.
21) Since equation of hyperbola is x2/a2 - y2/b2 =1 or x2/a2 + y2/(ib)2 =1 and that of an ellipse is x2/a2 + y2/b2 =1, hyperbolas can be perceived as an ellipse minor axis of which is imaginary.

HYPERBOLIC ANGLES

1) Angles are represented through circular functions because they satisfy  cos2θ + sin2θ =1 for all θ and the equation is analogous to the equation of a unit circle x2 + y2 =1 where (x,y)  are points on the circle and there exist θ that  satisfy x=cosθ & y= sinθ . Hence transformation from (x,y) --> (x',y') on  the circumference of a  circle are dependent on circular functions.

2) Similarly hyperbolic angles are represented through hyperbolic functions & because they satisfy cosh2θ - sinh2θ =1 for all hyperbolic θ and this equation is analogous to x2-y2=1 which is the equation of a unit hyperbola. So if (x,y)  are points on the hyperbola , there exist hyperbolic θ that  satisfy x=coshθ &  y= sinhθ and transformation from (x,y) --> (x',y') on  hyperbola  are dependent on hyperbolic functions.

3)We draw a unit circle and point P(a,b) and point P'(a,-b) on  C i,e circle .Measure of angle θ at P is arc length from x-axis to P along the circumference.arclength= value of area bounded by OP,OP' and arc of circle.
4) Extending it to unit hyperbola,we draw  point P(a,b) and point P'(a,-b) on  H i,e hyperbola.Measure of hyperbolic angle at P is arc length from x-axis to P along the circumference.arclength= value of area bounded by OP,OP' and arc of hyperbola..
5) Unlike circle which has a bounded area and measure of angle bounded to [-1,1], hyperbola has area unbounded and hyperbolic angle can take any non-negative real value and therefore for every real value r≥ 0,there is a unique hyperbolic angle whose measure is r.

Let b≥ 0. Let us draw a perpendicular from P to x-axis intersecting the axis at Q.Let it be l.
Let R be the intersection of Hyperbola H and x-axis.Draw a line through R so that it is perpendicular to x-axis & let S be the intersection point.

Sinhθ =PQ (red)coshθ =OQ(blue) and tanhθ=RS(green) since RS/PQ =OR/OQ or RS = OR*PQ / OQ =1*sinhθ / coshθ=tanhθ

(Trigonometric functions & Hyperbolic functions when x is a real number)

Trigonometric functions sin and cos are periodic functions bound by [-1,1] . But cosh functions are not less than 1. Moreover both cosh and sinh functions do not have any lower or upper bound.These are true for all real values of x. But if we take x along with  an imaginary number i, then we can correlate trigonometric and hyperbolic functions as under, the hyperbolic sin and cos functions become periodic like their trigonometric counterparts. Other formula, click

  coshx =  cosix

   sinhx = - isinix

While trigonometric ratios are defined with respect to the 3 sides of a right-angled triangle and the angles other than the right angle,  definition of hyperbolic  ratios are defined to what analogous yhings?