Equation of Hyperbola

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Hyperbola
A hyperbola is the set of all points in a plane, the difference
of whose distances from two fixed points in the plane is a
constant.
A hyperbola is a curve where the distances of any point
from:
– a fixed point (the focus), and
– a fixed straight line (the directrix) are always in the same ratio.
focus
Directrix
These distance
are always in
same ratio.
focus
Directrix
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 The two fixed points are called
the foci of the hyperbola.
 The mid-point of the line
segment joining the foci is
called the centre of the
hyperbola.
 The line through the foci is
called the transverse axis and
the line through the centre and
perpendicular to the transverse
axis is called the conjugate axis.
 The points at which the
hyperbola intersects the
transverse axis are called the
vertices of the hyperbola.
focus
Directrix
focus
Directrix
vertex vertex
conjugate axis
Transverse
axis Centre
The "asymptotes“ are not part of the
hyperbola, but show where the curve
would go if continued indefinitely in
each of the four directions.

 We denote the distance between the two foci by 2c, the
distance between two vertices by 2a and we define the
quantity b as
Also 2b is the length of
the conjugate axis
To find the constant
PF – PG:
By taking the point P at A
and B in the, we have
BF – BG = AG – AF (by the definition of the hyperbola)
BA + AF – BG = AB + BG – AF
i.e., AF = BG ,
so that , BF – BG = BA + AF – BG = BA = 2a
22
acb
2c
2a
a
XX’
Y
Y’
F GA B
b c

Standard equation of Hyperbola
• The equation of a hyperbola is simplest if the centre of the
hyperbola is at the origin and the foci are on the x-axis or
y-axis.
– We will derive the equation for the hyperbola shown in
with foci on the x-axis.
Let F and G be the foci and O
be the mid-point of the line
segment FG. Let O be the origin
and the line through O through G
be the positive x-axis and that
through F , as the negative x
axis. The line through O to the
x-axis be the y-axis.
G(c , 0)F(-c , 0 )
P (x , y)
x=-a
x=a
O
XX’
Y
Y’

Let the coordinates of F be (–c, 0) and G be (–c, 0) .
Let P(x , y) be any point on the hyperbola such that the
difference of the distances from P to the farther point
minus the closer point be 2a.
So given, PF – PG = 2a
Using the distance formula, we have
Squaring both side we get
square binomials
aycxycx 2)()( 2222
2222
)(2)( ycxaycxor
2222222
)(4)(4)( ycxaycxaycx
2222222
)(4242 ycxaxccxaxccx
22
)( ycxa
a
xc
or

By Squaring
i.e. ,
Hence any point on the hyperbola satisfies
22
2
)( ycxa
a
xc
2222
2
22
22 yxccxxca
a
cx
or
222
2
222
)(
acy
a
acx
or
1
)(
)(
22
2
222
222
ac
y
aca
acx
or
12
2
2
2
b
y
a
x
(Since b2 = c2 – a2)
12
2
2
2
b
y
a
x

Conversely, let P( x, y) satisfy the above equation with 0 < a< c.
Then,
Therefore,
Similarly,
2
22
22
a
ax
by
22
)( ycxPF
2
22
22
)(
a
ax
bcx
2
22
222
)()(
a
ax
accx
(Since b2 = c2 – a2)x
a
c
aPF
x
c
a
aPG

In hyperbola c> a; and since P is to the right of the line
x= a, x> a,
Therefore,
becomes negative. Thus,
Therefore,
Also, note that if P is to the left of the line x = –a, then
In that case PF – PG = 2 a
So, any point that satisfies lies on hyperbola
Thus, we proved that the equation of hyperbola with origin
(0,0) and transverse axis along x-axis is
.ax
a
c
x
c
a
a ax
a
c
PG
aax
a
c
x
a
c
aPGPF 2
,x
a
c
aPF x
c
a
aPG
12
2
2
2
b
y
a
x
12
2
2
2
b
y
a
x

222
2
2
2
2
where,1 acb
b
y
a
x
The equation for a hyperbola can be derived by using
the definition and the distance formula. The resulting
equation is:
aa
cb
b
This looks similar to the ellipse equation but notice the sign difference.
To graph a hyperbola, make a
rectangle that measures 2a by
2b as a sketching aid and
draw the diagonals. These
are the asymptotes.

Standard equation of a hyperbola with its center at
the origin and vertical transverse axis
For a hyperbola with its
center at the origin and
has the transverse axis
horizontal, the standard
equation is:
c2 = a2 + b2
2 2
2 2
1
y x
a b
The equations of its
asymptotes are:
x
a
y
b
Focus: (0, c) (0, -c) (0, c)
vertices: (0, a)(0, -a) (0, a)
(0, -c)
(0, c)
(0, -a)
(0, a)

The center of the hyperbola may be transformed
from the origin. The equation would then be:
horizontal
transverse
axis
12
2
2
2
b
ky
a
hx
12
2
2
2
b
hx
a
ky
vertical
transverse
axis
The axis is determined by the first term NOT by which
denominator is the largest. If the x term is positive it will be
horizontal, if the y term is the positive term it will be vertical.

Eccentricity
The eccentricity (usually shown as the letter e), it shows how
"uncurvy" (varying from being a circle) the hyperbola is.
On this diagram:
 P is a point on the curve,
 F is the focus and
 N is the point on the directrix so
that PN is perpendicular to the
directrix.
The ratio PF/PN is the eccentricity of the hyperbola (for a
hyperbola the eccentricity is always greater than 1).
It can also given by the formula:
F
PN
Directrix
a
ba
e
22

Latus rectum of Hyperbola
Latus rectum of hyperbola is a line segment perpendicular
to the transverse axis through any of the foci and whose
end points lie on the hyperbola.
Directrix
F
Latus rectum
The length of the Latus Rectum is
a
b2
2

The End
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Equation of Hyperbola

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