Formula Master Part I

CHAPTER 11
CONIC SECTION

1. Locus of a point: When a point moves in such a way that its path traced out is
called its locus. E.g.: sector of a circle, circle, conic, etc.

2. A conic is a curve obtained by slicing a cone with a plane which does not pass
through the vertex. It is also known as conic section.

3. Types of conics:

i. Parabola : Parabola is a conic obtained by slicing a cone with a plane,

which does not pass through the vertex and parallel to any
generators.
ii. Ellipse : Ellipse is a conic obtained by slicing a cone with a plane, which
does not pass through the vertex not parallel to any generators
and cuts only one nappe.
iii. Hyperbola : Hyperbola is a conic obtained by slicing a cone with a plane,
which does not pass through the vertex not parallel to any
generators and cuts two nappes.
iv. Circle : Circle is also a conic obtained by slicing a cone with a plane,
which does not pass through the vertex, parallel to its base.

axis generator

ellipse
Upper nappe
vertex

hyperbola circle
Lower nappe
generator

parabola

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4. Conic section as a locus: The conic section is the locus of a point which moves in a
plane so that its distance from a fixed point bears a constant ratio to its distance
from a fixed line.
5. The fixed point is called focus (S) and the fixed line is called directrix and the
constant ratio is called eccentricity of the conic, which is denoted by e.
6. If e=1 the conic is known as a parabola, if e  1 , the conic is known as an ellipse
and if e  1 , the conic is known as a hyperbola and if e  0 , the conic is known as a
circle.

PARABOLA

Standard Equation
Four types of parabolas:

Name of the Right handed Left handed Upward Downward

parabola Parabola Parabola Parabola Parabola

Graph

Standard form y 2  4ax y 2  4ax x 2  4ay x 2  4ay

Vertex A  0, 0  A  0, 0  A  0, 0  A  0, 0 

Focus S  a, 0  S  a, 0  S  0, a  S  0, a 

Equation of the x  a xa y  a ya

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directrix
Axis x axis x axis y axis y axis

Symmetry x axis x axis y axis y axis

Equation of the axis y0 y0 x0 x0

Equation of the x0 x0 y0 y0

tangent at the vertex
Length of the latus 4a 4a 4a 4a
rectum
Equation of the latus xa x  a ya y  a
rectum

ELLIPSE

Definition2: An ellipse is the set of all points in the plane whose distances from a fixed
point in the plane bears a constant ratio, less than one, to their distances
from a fixed line in the plane.
Definition3: An ellipse the set of all points in a plane, the sum of whose distances from
two fixed points in the plane is a constant.
Definition4: If e<1, then the conic is known as an ellipse.

MAJOR AXIS IS ALONG THE X-AXIS:

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MAJOR AXIS IS ALONG THE Y-AXIS

PROPERTIES

Major axis is along Major axis is along the

Name of the Ellipse
the x axis y axis
Standard form x2 y2 x2 y2
  1 , a 2  b2   1, a 2  b 2
a2 b2 b2 a2
Centre C  0, 0  C  0, 0 

Foci S(ae, 0),S( ae, 0) S(0, ae),S(0, ae)

Vertices A  a, 0  , A   a, 0  A  0, a  , A   0, a 

Equation of the directrix a a

x y
e e
Equation of the major axis y0 x0
Equation of the minor axis x0 y0

Length of latus rectum 2b 2 2b 2

a a

Equation of the latus x  ae y   ae

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rectum
Length of major axis 2a 2a
Length of minor axis 2b 2b

Note:


1. b 2  a 2 1  e 2 
2. Centre of the ellipse is the point of intersection of the major and minor axis.
3. Foci are the point of intersection of major axis and the latus recta.
4. The ratio of the distances from the centre of the ellipse to one of the foci and to one
of the vertices of the ellipse is known as eccentricity.
c
The eccentricity, e  , where c  a 2  b 2
a

HYPERBOLA

Definition2: A hyperbola is the set of all points in the plane whose distance from a
fixed point in the plane bears a constant ratio, greater than one, to their
distances from a fixed line in the plane.
Definition3: A hyperbola the set of all points in a plane, the sum of whose distances
from two fixed points in the plane is a constant.
Definition4: If e>1, then the conic is known as a hyperbola.

TRANSVERSE AXIS IS ALONG THE X AXIS

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TRANSVERSE AXIS IS ALONG THE Y AXIS

PROPERTIES

Transverse axis is along Transverse axis is along

Name of the hyperbola
the x axis the y axis
Standard form x2 y2 y2 x2
 1  1
a2 b2 a2 a2
Centre C  0, 0  C  0, 0 

Foci S(ae, 0),S( ae, 0) S(0, ae),S(0, ae)

Vertices A  a, 0  , A   a, 0  A  0, a  , A   0, a 

Equation of the directrix a a

x y
e e
Equation of the y0 x0
transverse axis
Equation of the x0 y0
conjugate axis

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Length of latus rectum 2b 2 2b 2

a a

Equation of the latus x  ae y   ae

rectum
Length of transverse 2a 2a
axis
Length of conjugate axis 2b 2b
Equation of the y0 x0
transverse axis
Equation of the x0 y0
conjugate axis

Note:

1. 
b2  a 2 e2  1 
2. Centre of the hyperbola is the point of intersection of the transverse axis and the
conjugate axis.
3. Foci are the point of intersection of transverse axis and the latus recta.
4. The ratio of the distances from the centre of the hyperbola to one of the foci and to
one of the vertices of
5. the hyperbola is known as eccentricity.
c
The eccentricity, e  , where c  a 2  b 2
6. a

7. Hyperbola in which a=b is called equilateral hyperbola.

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