Hyperbola
A hyperbola is the locus of a point in a plane which moves in the plane
in such a way that the ratio of its distance from a fixed point in the
same plane to its distance from a fixed line is always constant, which is
always greater than unity.
The fixed point is called the focus and the fixed line is directrix and
the ratio is the eccentricity.
Transverse and Conjugate Axes
(i) The line through the foci of the hyperbola is called its transverse
axis.
(ii) The line through the centre and perpendicular to the transverse
axis of the hyperbola is called its conjugate axis.
x2 y2
Hyperbola of the Form − =1
a2 b2
Y
L′ L
P
S1 A1
X' A S
X
O
L′1 l' l L1
Y'
(i) Centre : O( 0, 0)
(ii) Foci : S ( ae, 0), S1( − ae, 0)
(iii) Vertices : A( a , 0), A1 ( − a , 0)
a ′ a
(iv) Equation of directrices l : x = ,l : x = −
e e
2 b2
(v) Length of latusrectum : LL 1 = L ′ L ′1 =
a
(vi) Length of transverse axis : 2a
, (vii) Length of conjugate axis : 2b
2
b
(viii) Eccentricity e = 1 +
a
or b2 = a 2( e2 − 1)
(ix) Distance between foci = 2ae
2a
(x) Distance between directrices =
e
b2
(xi) Coordinates of ends of latusrectum = ± ae, ±
a
(xii) Focal radii|SP | =|ex1 − a| and|S1 P| =|ex1 + a|
x2 y2
Conjugate Hyperbola – + =1
a2 b2
(i) Centre : O( 0, 0) Y
S
(ii) Foci : S ( 0, be ), S1 ( 0, − be) L1 L
P
(iii) Vertices : A ( 0, b), A1( 0, − b) A
l
(iv) Equation of directrices
b b X' X
l : y = ,l ′ : y = − O
e e l'
A1
(v) Length of latusrectum :
L'1 L'
2a 2 S1
LL1 = L ′ L1 ′ =
b Y'
(vi) Length of transverse axis : 2b.
(vii) Length of conjugate axis : 2a.
2
a
(viii) Eccentricity e = 1 +
b
(ix) Distance between foci = 2be
2b
(x) Distance between directrices =
e
a2
(xi) Coordinates of ends of latusrectum = ± , ± be
b
(xii) Focal radii|SP | =|ey1 − b| and|S1P| =|ey1 + b|
A hyperbola is the locus of a point in a plane which moves in the plane
in such a way that the ratio of its distance from a fixed point in the
same plane to its distance from a fixed line is always constant, which is
always greater than unity.
The fixed point is called the focus and the fixed line is directrix and
the ratio is the eccentricity.
Transverse and Conjugate Axes
(i) The line through the foci of the hyperbola is called its transverse
axis.
(ii) The line through the centre and perpendicular to the transverse
axis of the hyperbola is called its conjugate axis.
x2 y2
Hyperbola of the Form − =1
a2 b2
Y
L′ L
P
S1 A1
X' A S
X
O
L′1 l' l L1
Y'
(i) Centre : O( 0, 0)
(ii) Foci : S ( ae, 0), S1( − ae, 0)
(iii) Vertices : A( a , 0), A1 ( − a , 0)
a ′ a
(iv) Equation of directrices l : x = ,l : x = −
e e
2 b2
(v) Length of latusrectum : LL 1 = L ′ L ′1 =
a
(vi) Length of transverse axis : 2a
, (vii) Length of conjugate axis : 2b
2
b
(viii) Eccentricity e = 1 +
a
or b2 = a 2( e2 − 1)
(ix) Distance between foci = 2ae
2a
(x) Distance between directrices =
e
b2
(xi) Coordinates of ends of latusrectum = ± ae, ±
a
(xii) Focal radii|SP | =|ex1 − a| and|S1 P| =|ex1 + a|
x2 y2
Conjugate Hyperbola – + =1
a2 b2
(i) Centre : O( 0, 0) Y
S
(ii) Foci : S ( 0, be ), S1 ( 0, − be) L1 L
P
(iii) Vertices : A ( 0, b), A1( 0, − b) A
l
(iv) Equation of directrices
b b X' X
l : y = ,l ′ : y = − O
e e l'
A1
(v) Length of latusrectum :
L'1 L'
2a 2 S1
LL1 = L ′ L1 ′ =
b Y'
(vi) Length of transverse axis : 2b.
(vii) Length of conjugate axis : 2a.
2
a
(viii) Eccentricity e = 1 +
b
(ix) Distance between foci = 2be
2b
(x) Distance between directrices =
e
a2
(xi) Coordinates of ends of latusrectum = ± , ± be
b
(xii) Focal radii|SP | =|ey1 − b| and|S1P| =|ey1 + b|
