Ellipse
Ellipse is the locus of a point in a plane which moves in such a way
that the ratio of the distance from a fixed point (focus) in the same
plane to its distance from a fixed straight line (directrix) is always
constant, which is always less than unity.
Major and Minor Axes
The line segment through the foci of the ellipse with its end points on
the ellipse, is called its major axis.
The line segment through the centre and perpendicular to the major
axis with its end points on the ellipse, is called its minor axis.
x2 y2
Horizontal Ellipse i.e. + = 1,( 0 < b < a )
a2 b2
If the coefficient of x 2 has the larger denominator, then its major axis
lies along the X-axis, then it is said to be horizontal ellipse.
Z′ Y Z
B(0, b)
P(x, y)
M
S′(–ac, 0) K
X′ X
K′ A′(–a, 0) C S N A
(ae, 0) (a,0)
B′ (0, –b) a
a x=
x=– e
e
(i) Vertices A( a , 0), A1( − a , 0)
(ii) Centre O ( 0, 0)
(iii) Length of major axis, AA1 = 2a; Length of minor axis, BB1 = 2b
(iv) Foci are S ( ae , 0) and S1( − ae , 0)
a a
(v) Equation of directrices are l : x = ,l ′ ; x = −
e e
, 2b2
(vi) Length of latusrectum, LL1 = L ′ L1 ′ =
a
b2
(vii) Eccentricity, e = 1 − <1
a2
(viii) Focal distances of point P ( x , y ) are SP and S1P i.e.|a − ex| and
|a + ex|. Also, SP + S1P = 2a = major axis.
(ix) Distance between foci = 2ae
2a
(x) Distance between directrices =
e
x2 y2
Vertical Ellipse i.e. + = 1, ( 0 < a < b )
a2 b2
If the coefficient of x 2 has the smaller denominator, then its major axis
lies along the Y -axis, then it is said to be vertical ellipse.
Y
B l
L1 S L
N
P' P(x,y)
X' X
A1 O A
S1 L'
L'1
B1 l'
Y'
(i)Vertices B ( 0, b), B1( 0, − b)
(ii)Centre O( 0, 0)
(iii)Length of major axis BB1 = 2b, Length of Minor axis AA1 = 2a
(iv) Foci are S ( 0, ae ) and S1( 0, − ae)
b b
(v) Equation of directrices are l : y = ; l ′ : y = −
e e
2a 2
(vi) Length of latusrectum LL1 = L ′ L1 ′ =
b
a2
(vii) Eccentricity e = 1 − <1
b2
Ellipse is the locus of a point in a plane which moves in such a way
that the ratio of the distance from a fixed point (focus) in the same
plane to its distance from a fixed straight line (directrix) is always
constant, which is always less than unity.
Major and Minor Axes
The line segment through the foci of the ellipse with its end points on
the ellipse, is called its major axis.
The line segment through the centre and perpendicular to the major
axis with its end points on the ellipse, is called its minor axis.
x2 y2
Horizontal Ellipse i.e. + = 1,( 0 < b < a )
a2 b2
If the coefficient of x 2 has the larger denominator, then its major axis
lies along the X-axis, then it is said to be horizontal ellipse.
Z′ Y Z
B(0, b)
P(x, y)
M
S′(–ac, 0) K
X′ X
K′ A′(–a, 0) C S N A
(ae, 0) (a,0)
B′ (0, –b) a
a x=
x=– e
e
(i) Vertices A( a , 0), A1( − a , 0)
(ii) Centre O ( 0, 0)
(iii) Length of major axis, AA1 = 2a; Length of minor axis, BB1 = 2b
(iv) Foci are S ( ae , 0) and S1( − ae , 0)
a a
(v) Equation of directrices are l : x = ,l ′ ; x = −
e e
, 2b2
(vi) Length of latusrectum, LL1 = L ′ L1 ′ =
a
b2
(vii) Eccentricity, e = 1 − <1
a2
(viii) Focal distances of point P ( x , y ) are SP and S1P i.e.|a − ex| and
|a + ex|. Also, SP + S1P = 2a = major axis.
(ix) Distance between foci = 2ae
2a
(x) Distance between directrices =
e
x2 y2
Vertical Ellipse i.e. + = 1, ( 0 < a < b )
a2 b2
If the coefficient of x 2 has the smaller denominator, then its major axis
lies along the Y -axis, then it is said to be vertical ellipse.
Y
B l
L1 S L
N
P' P(x,y)
X' X
A1 O A
S1 L'
L'1
B1 l'
Y'
(i)Vertices B ( 0, b), B1( 0, − b)
(ii)Centre O( 0, 0)
(iii)Length of major axis BB1 = 2b, Length of Minor axis AA1 = 2a
(iv) Foci are S ( 0, ae ) and S1( 0, − ae)
b b
(v) Equation of directrices are l : y = ; l ′ : y = −
e e
2a 2
(vi) Length of latusrectum LL1 = L ′ L1 ′ =
b
a2
(vii) Eccentricity e = 1 − <1
b2
