Conic Sections
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Conic Sections
Conic Sections Standard Form
A conic (section) is the locus of a point moving in a plane, Important results of a parabola
such that its distance from a fixed point (focus) is in a 1. 4 x distance between vertex and focus = Latus rectum =
constant ratio to its perpendicular distance from a fixed 4a.
line, i.e., directrix. This constant ratio is called the 2. 2 x distance between directrix and focus = Latus rectum
eccentricity of the conic. = 2(2a).
The eccentricity of a circle is zero. It shows how “un- 3. Point of intersection of the axis and directrix, and the
circular” a curve is. Higher the eccentricity, the lower curved focus is bisected by the vertex.
it is. 4. Focus is the midpoint of the Latus rectum.
5. (Distance of any point on the parabola from the axis)2 =
(LR) (Distance of the same point from tangent at the vertex)
Ellipse
It is a locus of a point which moves such that the ratio of its
distance from a fixed point (focus) to its distance from a
fixed line (directrix) is always constant and less than 1, i.e., 0
< e < 1.
Terminology:
Axis of conic: Line passing through focus, perpendicular to
the directrix.
Vertex: Point of the intersection of conic and axis.
Chord: Line segment joining any 2 points on the conic.
Double ordinate: Chord perpendicular to the axis
Latus rectum: Double ordinate passing through focus.
Standard Parabola
Length
Standard
Directrix Focus of Latus Vertex Ellipse with a horizontal major axis
Equation
Rectum
x2 y 2
+ = 1;b a
a2 b2
y 2 = 4ax x=-a
( )
Focus: There are 2 focii: ae,0 and −ae,0 ( )
S : ( a,0 ) 4a ( 0,0 ) Directrix: These foci have corresponding directrices as
x = +a / e and x = −a / e , respectively.
b2
e2 = 1 −
y = −4ax
2
x=a a2
( −a,0 ) 4a ( 0,0 ) Axes:
xx': Major Axis: Length: 2a
yy : Minor axis: length : 2b
x2 = 4ay y=-a
( 0,0 ) ( ) (
Vertex: a,0 & −a,0 )
( 0, +a) 4a
Centre: ( 0,0 )
Latus rectum:
x = −4ay
2
y=a b2
( 0, −a) 4a ( 0,0 ) y=
R
x2 y 2
(solve x = ae with + =1 )
a2 b2
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Conic Sections
Length of Latus rectum: Hyperbola
2b2 It is the locus of a point which moves such that the ratio of
y= its distance from a fixed point (focus) to its distance from a
a
fixed line (directrix) is always constant and greater than 1.
Ellipse with the vertical major axis
e>1
x2 y 2
+ = 1;b a
a2 b2
a2
e2 = 1 − 2
b
Length of major axis: 2b
Length of minor axis: 2a
( ) (
Focii: 0,be & 0, −be )
Directrices: x = b / e & x = −b / e
Latus rectum: y = + be Standard hyperbola:
Length of Latus rectum: Equation:
2a 2 x2 y 2
− =1
b a2 b2
Important results Focus: There are 2 foci (ae, 0) and (-ae, 0)
( semi minor axis)
1. e = 1 − Directrix: The foci have corresponding directrices as x =
( semi major axis )2 +a/e and x = −a/e, respectively.
2( semi minor axis )2
2. Length of Latus Rectum = Axis: xx’ : Transverse axis ; Length : 2a
( semi minor axis ) yy’ : Conjugate axis ; Length : 2b (Hypothetical)
Major axis
3. Distance between 2 directices:
eccentricity Vertex : (0, 0) and (-a, 0)
4. Distance between 2 foci: (major axis) × eccentricity
Centre: (0, 0)
a b
5. Distance between focus and directrix: − ae or − be
e e Latus rectum: x = +ae
Area of ellipse = ab
Where, Length of latus rectum:
• a = length of semi-major axis 2b2a
• b = length of semi-minor ax is
Position of point at hyperbola
Area of an Ellipse Formula
x2a2−y2b2=1
Let S=x2a2−y2b2−1
and
S1=x12a2−y12b2−1
If S1 > 0, point C lies inside the hyperbola
S1 = 0 point B lies on the hyperbola
S1 < 0 point A lies outside the hyperbola.
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Conic Sections
In Short:
Equations of some of the conic sections when the centre
is the origin or any given point, say (h, k), are as follows:
Conic
section
Centre at origin ( )
Centre is h,k
x2 + y2 = r2 ;r (x − h)2 + (y − k)2
Circle = r 2 ;r
is the radius is the radius
(x 2
)
/ a2 − (x − h)2 / a2 −
Hyperbola
(y 2
/b ) =1
2
(y − k)2 / b2 = 1
Ellipse
(x 2
) (
/ a2 + y 2 / b2 ) (x − h)2 / a2 +
=1 (y − k)2 / b2 = 1
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