Four standard forms of hyperbola:
Content I II III IV
2. Centre(C) (0,0) (0,0) (α,β) (α,β)
3. Vertices A,A’=(±a,0 ) B,B’=(0,±b) (α±a, β) (α,b± β)
4 . Foci(S,S’) (±ae,0) (0,±be) (α±ae,β) (α,β±be)
5. Z,Z’ (±a/e,0) (0,±b/e) (α±a/e,β) (α,β±b/e)
focal distance(focal radii)
of a point P on the Hyperbola
II. The hyperbola x2/a2-y2/b2=1 and x2/a2-y2/b2=-1 are called conjugate hyperbolas to each other .
III. If e1,e2 are the two eccentricities of two conjugate hyperbolas then e12+ e22= e12 e22
IV. A hyperbola is said to be a rectangular hyperbola if the length of its transverse axis is equal to length of its conjugate axis
V. x2-y2=a2,xy=c2 represents a rectangular hyperbola
VI. The eccentricity of a rectangular hyperbola is
VII. We use the following notation in this chapter
IX.A point is said (x1,y1) is said to be 0:-an external point to the hyperbola S=0 if S11<0.
X. Two tangents can be drawn to a hyperbola from an external point.
XI. The equation of the tangent to a hyperbola S=0 at P(x1,y1) is S1=0
XII. The equation of the normal to the hyperbola x2/a2-y2/b2=1 at P(x1,y1) is a2x/x1 + b2y/b1=a2+b2
XIV. The condition that the line y=mx+c may be a tangent to the hyperbola x2/a2-y2/b2=1 is c2=a2m2-b2 and the point of contact is (-a2m/c , -b2/c)
XV. The condition that the line lx+my+n=0 may be a tangent to the hyperbola x2/a2-y2/b2=1 is (-a2l/n,-b2m/n).
XVI. The equation of a tangent to the hyperbola x2/a2-y2/b2=1 may be taken as y=mx±
XVII. If m1,m2 are the slopes of the tangents through P to the hyperbola x2/a2-y2/b2=1 then m1+m2=2x1y1/x12-a2 ; m1m2=y12+ b2/ x12-a2
XIX. The equation to the director circle x2/a2-y2/b2=1 is x2+y2=a2-b2
XX. The equation to the auxiliary circle of x2/a2-y2/b2=1 is x2+y2=a2
XXI. The equation to the chord of contact of P(x1,y1) with respect to the hyperbola S=0 is S1=0
XXII. The qquation to the chord of the hyperbola S=0 having P(x1,y1) as its midpoint is S1=S12
XXIV. The midpoint of the chord lx+my+n=0 of the hyperbola x2/a2-y2/b2=1 is (-a2ln/ a2l2-b2 m2,b2mn/ a2l2-b2 m2 )
XXV. The equation to the pair of tangents to the hyperbola S=0 from P(x1,y1) is S12=SS11
XXVI. The equation x=asec
XXVII. If P(x1,y1) =(asec
XXVII. The equation of the chord joining two points a and b on the hyperbola x2/a2-y2/b2=1 is x/a cos(α-β/2)-y/bsin(α+ β/2)=Cos (α+ β/2)
XXX. If a and b are the ends of a focal chord of a hyperbola S=0 then e cos(α- β/2)= Cos (α+ β/2)
XXXI. The equation of tangent at p(
XXXII. The equation of normal at p(
XXXIII. The condition that the line lx+my+n=0 to be a normal to the hyperbola x2/a2-y2/b2=1 is a2/ l2-b2/m2=( a2+ b2)2/n2
XXXIV. Atmost four normals can be drawn from a point to a hyperbola
XXXV. The parametric equation of xy=c2 are x=ct;y=c/t
Properties of Asymptotes:
ii). The equation to the pair of asymptotes of x2/a2-y2/b2=1 is x2/a2-y2/b2=0
iii). Equation of hyperbola and equation of its pair of asymptotes are differ in their constant terms only.
iv). Asymptotes of the hyperbola passes through the centre of the hyperbola and they are equally inclined to the axes of the hyperbola
v). The angle between the asymptotes of the hyperbola S=0 is 2
vi). The angle between the asymptotes of a rectangular hyperbola is
vii). The equation of a rectangular hyperbola whose asymptotes are the coordinate axes is xy=c2
viii). The product of perpendiculars from any point on hyperbola S=0 to its asymptotes is a2b2/a2+b2
ix). Asymptotes of a hyperbola and its conjugate hyperbola are same.
x). If H,C and A are the equation of a hyperbola and its pair of asymptotes respectively then H+C=2A
xi). Equation of pair of asymptotes of hyperbola ax2+2hxy+by2+2gx+2fy+c=0 is ax2+2hxy+by2+2gx+2fy+c-