Page 1
KEY POINTS
? Circle, ellipse, parabola and hyperbola are curves which are
obtained by intersection of a plane and cone in different
positions
? Circle:It is the set of all points in a plane that are equidistant
from a fixed point in that plane
? Equation of circle: (x – h)
2
+ (y – k)
2
= r
2
Centre (h, k), radius = r
? Parabola:It is the set of all points in a plane which are
equidistant from a fixed point (focus) and a fixed line (directrix)
in the plane. Fixed point does not lie on the line.
Page 2
KEY POINTS
? Circle, ellipse, parabola and hyperbola are curves which are
obtained by intersection of a plane and cone in different
positions
? Circle:It is the set of all points in a plane that are equidistant
from a fixed point in that plane
? Equation of circle: (x – h)
2
+ (y – k)
2
= r
2
Centre (h, k), radius = r
? Parabola:It is the set of all points in a plane which are
equidistant from a fixed point (focus) and a fixed line (directrix)
in the plane. Fixed point does not lie on the line.
Main facts about the Parabola
Equation y
2
= 4 a x y
2
= –4 a x x
2
= 4 a y x
2
= –4 a y
(a > 0) a > 0 a > 0 a > 0
Right hand Left hand Upwards Downwards
Axis y = 0 y = 0 x = 0 x = 0
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0
Focus (a, 0) (–a, 0) (0, a) (0, –a)
Length of
latus–rectum
4a 4a 4a 4a
Equation of
latus–rectum
x – a = 0 x + a = 0 y – a = 0 y + a = 0
? Latus Rectum:A chord through focus perpendicular to axis of
parabola is called its latus rectum.
O
,
X´
Y
y = 4ax
2
x
+
a
=
0
F(–a,0)
O
Y
x
–
a
=
0
X´
,
y = –4ax
2
F(a,0)
O
,
Y
x = 4ay
2
F(0, a)
X
y + a = 0
O
Y
x = –4ay
2
F
(
0
,
–
a
)
X
y – a = 0
Page 3
KEY POINTS
? Circle, ellipse, parabola and hyperbola are curves which are
obtained by intersection of a plane and cone in different
positions
? Circle:It is the set of all points in a plane that are equidistant
from a fixed point in that plane
? Equation of circle: (x – h)
2
+ (y – k)
2
= r
2
Centre (h, k), radius = r
? Parabola:It is the set of all points in a plane which are
equidistant from a fixed point (focus) and a fixed line (directrix)
in the plane. Fixed point does not lie on the line.
Main facts about the Parabola
Equation y
2
= 4 a x y
2
= –4 a x x
2
= 4 a y x
2
= –4 a y
(a > 0) a > 0 a > 0 a > 0
Right hand Left hand Upwards Downwards
Axis y = 0 y = 0 x = 0 x = 0
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0
Focus (a, 0) (–a, 0) (0, a) (0, –a)
Length of
latus–rectum
4a 4a 4a 4a
Equation of
latus–rectum
x – a = 0 x + a = 0 y – a = 0 y + a = 0
? Latus Rectum:A chord through focus perpendicular to axis of
parabola is called its latus rectum.
O
,
X´
Y
y = 4ax
2
x
+
a
=
0
F(–a,0)
O
Y
x
–
a
=
0
X´
,
y = –4ax
2
F(a,0)
O
,
Y
x = 4ay
2
F(0, a)
X
y + a = 0
O
Y
x = –4ay
2
F
(
0
,
–
a
)
X
y – a = 0
? Ellipse: It is the set of points in a plane the sum of whose
distances from two fixed points in the plane is a constant
and is always greater than the distances between the
fixed points
Main facts about the ellipse
Equation
2 2
2 2
1
x y
a b
? ?
2 2
2 2
1
x y
b a
? ?
a > 0, b > 0 a > 0, b >
0
Centre (0,0) (0,0)
Major axis lies along x–axis y–axis
Length of major axis 2a 2a
Length of minor axis 2b 2b
Page 4
KEY POINTS
? Circle, ellipse, parabola and hyperbola are curves which are
obtained by intersection of a plane and cone in different
positions
? Circle:It is the set of all points in a plane that are equidistant
from a fixed point in that plane
? Equation of circle: (x – h)
2
+ (y – k)
2
= r
2
Centre (h, k), radius = r
? Parabola:It is the set of all points in a plane which are
equidistant from a fixed point (focus) and a fixed line (directrix)
in the plane. Fixed point does not lie on the line.
Main facts about the Parabola
Equation y
2
= 4 a x y
2
= –4 a x x
2
= 4 a y x
2
= –4 a y
(a > 0) a > 0 a > 0 a > 0
Right hand Left hand Upwards Downwards
Axis y = 0 y = 0 x = 0 x = 0
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0
Focus (a, 0) (–a, 0) (0, a) (0, –a)
Length of
latus–rectum
4a 4a 4a 4a
Equation of
latus–rectum
x – a = 0 x + a = 0 y – a = 0 y + a = 0
? Latus Rectum:A chord through focus perpendicular to axis of
parabola is called its latus rectum.
O
,
X´
Y
y = 4ax
2
x
+
a
=
0
F(–a,0)
O
Y
x
–
a
=
0
X´
,
y = –4ax
2
F(a,0)
O
,
Y
x = 4ay
2
F(0, a)
X
y + a = 0
O
Y
x = –4ay
2
F
(
0
,
–
a
)
X
y – a = 0
? Ellipse: It is the set of points in a plane the sum of whose
distances from two fixed points in the plane is a constant
and is always greater than the distances between the
fixed points
Main facts about the ellipse
Equation
2 2
2 2
1
x y
a b
? ?
2 2
2 2
1
x y
b a
? ?
a > 0, b > 0 a > 0, b >
0
Centre (0,0) (0,0)
Major axis lies along x–axis y–axis
Length of major axis 2a 2a
Length of minor axis 2b 2b
Foci (–c, 0), (c, 0) (0, –c),(0, c)
Vertices (–a, 0), (a, 0) (0, –a), (0, a)
Eccentricity (e)
Length of latus–rectum
c
a
2
2 b
a
c
a
2
2 b
a
? If e = 0 for ellipse than b = a and equation of ellipse will be
converted in equation of the circle its eq. will be x² + y² = a². it is
called auxiliary circle. For auxiliary circle, diameter is equal to
length of major axis and e = 0
? Latus rectum:Chord through foci perpendicular to major axis
called latus rectum.
? Hyperbola:It is the set of all points in a plane, the differences of
whose distance from two fixed points in the plane is a constant.
Page 5
KEY POINTS
? Circle, ellipse, parabola and hyperbola are curves which are
obtained by intersection of a plane and cone in different
positions
? Circle:It is the set of all points in a plane that are equidistant
from a fixed point in that plane
? Equation of circle: (x – h)
2
+ (y – k)
2
= r
2
Centre (h, k), radius = r
? Parabola:It is the set of all points in a plane which are
equidistant from a fixed point (focus) and a fixed line (directrix)
in the plane. Fixed point does not lie on the line.
Main facts about the Parabola
Equation y
2
= 4 a x y
2
= –4 a x x
2
= 4 a y x
2
= –4 a y
(a > 0) a > 0 a > 0 a > 0
Right hand Left hand Upwards Downwards
Axis y = 0 y = 0 x = 0 x = 0
Directrix x + a = 0 x – a = 0 y + a = 0 y – a = 0
Focus (a, 0) (–a, 0) (0, a) (0, –a)
Length of
latus–rectum
4a 4a 4a 4a
Equation of
latus–rectum
x – a = 0 x + a = 0 y – a = 0 y + a = 0
? Latus Rectum:A chord through focus perpendicular to axis of
parabola is called its latus rectum.
O
,
X´
Y
y = 4ax
2
x
+
a
=
0
F(–a,0)
O
Y
x
–
a
=
0
X´
,
y = –4ax
2
F(a,0)
O
,
Y
x = 4ay
2
F(0, a)
X
y + a = 0
O
Y
x = –4ay
2
F
(
0
,
–
a
)
X
y – a = 0
? Ellipse: It is the set of points in a plane the sum of whose
distances from two fixed points in the plane is a constant
and is always greater than the distances between the
fixed points
Main facts about the ellipse
Equation
2 2
2 2
1
x y
a b
? ?
2 2
2 2
1
x y
b a
? ?
a > 0, b > 0 a > 0, b >
0
Centre (0,0) (0,0)
Major axis lies along x–axis y–axis
Length of major axis 2a 2a
Length of minor axis 2b 2b
Foci (–c, 0), (c, 0) (0, –c),(0, c)
Vertices (–a, 0), (a, 0) (0, –a), (0, a)
Eccentricity (e)
Length of latus–rectum
c
a
2
2 b
a
c
a
2
2 b
a
? If e = 0 for ellipse than b = a and equation of ellipse will be
converted in equation of the circle its eq. will be x² + y² = a². it is
called auxiliary circle. For auxiliary circle, diameter is equal to
length of major axis and e = 0
? Latus rectum:Chord through foci perpendicular to major axis
called latus rectum.
? Hyperbola:It is the set of all points in a plane, the differences of
whose distance from two fixed points in the plane is a constant.
Main facts about the Hyperbola
Equation
2 2
2 2
1
x y
a b
? ?
2 2
2 2
1
x y
b a
? ?
a > 0, b > 0 a > 0, b > 0
Latus Rectum:Chord through foci perpendicular to transverse axis is
called latus rectum.
If 2 e ? for hyperbola, then hyperbola is called rectangular hyperbola.
For 2 e ? then b = a and eq. of its hyperbola will be
2 2 2
x y a ? ? or
2 2 2
y x a ? ? .
(A) Circle:If the equation of the circle is in the form of
2 2
2 2 0 x y gx fy c ? ? ? ? ?
then centre of circle of = c (–g, –f)
and radius of circle =
2 2
r g f c ? ? ?
(B) Parabola:in figure:-
(i) Equation of the parabola y² = 4ax.
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