This document discusses conic sections, including circles, ellipses, parabolas, and hyperbolas. It provides: 1) The definitions and standard equations of each conic section, describing how they are formed from the intersection of a plane with a double cone. 2) Examples of different forms the equations can take and the geometric properties of each conic section, such as foci, axes, vertices, and asymptotes. 3) Methods for writing the equations of tangents to conics and using parametric equations to represent loci. In less than 3 sentences, it summarizes the key information about conic sections provided in the document.

1. REVISITING……
Conic Sections

2. Conic Sections
(1) Circle
A circle is formed when


2
i.e. when the plane  is
perpendicular to the
axis of the cones.

3. Conic Sections
(2) Ellipse
An ellipse is formed when
  

2
i.e. when the plane  cuts
only one of the cones, but
is neither perpendicular to
the axis nor parallel to the
a generator.

4. Conic Sections
(3) Parabola
A parabola is formed when
 
i.e. when the plane  is
parallel to a generator.

5. Conic Sections
(4) Hyperbola
A hyperbola is formed when
0  
i.e. when the plane  cuts
both the cones, but does not
pass through the common
vertex.

6. CIRCLE
A circle is the locus of a variable point on a
plane so that its distance (the radius)remains
constant from a fixed point (the centre).
y
P(x,y)

O
x

7. DIFFERENT FORMS OF EQUATIONS OF CIRCLE

×
The standard equation of circle:
( x  h)2  ( y  k )2  r 2
where (h, k )is the centre of the circle and r is its radius.
The parametric equation of a circle:
x  r cos  ,
×
y  r sin 
The general equation of a circle:
x2  y 2  2 gx  2 fy  c  0
where
( g ,  f )
g2  f 2  c
is the centre of the circle and
is its radius …..

8. Parabola
A parabola is the locus of a variable point on a
plane so that its distance from a fixed point
(the focus) is equal to its distance from a fixed
line (the directrix x = - a).
y
P(x,y)
M(-a,0)
O
focus F(a,0)
x

9. Form the definition of parabola,
PF = PN
( x  a)  y  x  a
2
2
( x  a)  y  ( x  a)
2
2
2
x  2ax  a  y  x  2ax  a
2
2
2
2
2
y  4ax
2
standard equation of a parabola

10. vertex
axis of symmetry
latus rectum (LL’)
mid-point of FM = the origin (O) = vertex
length of the latus rectum =LL`= 4a

12. Other forms of Parabola
y  4ax
2

13. Other forms of Parabola
x  4ay
2

14. Other forms of Parabola
x  4ay
2

15. 12.1 Equations of a Parabola
A parabola is the locus of a variable point P
which moves in a plane so that its distance from
a fixed point F in the plane equals its distance
from a fixed line l in the plane.
The fixed point F is
called the focus and the
fixed line l is called the
directrix.

16. 12.1 Equations of a Parabola
The equation of a parabola with focus
F(a,0) and directrix x + a =0, where a >0,
is y2 = 4ax.

17. 12.1 Equations of a Parabola
X`X is the axis.
O is the vertex.
F is the focus.
MN is the focal chord.
HK is the latus rectum.

18. DIFFERENT FORMS OF EQUATIONS OF
PARABOLA

The standard equation of parabola:
2
( y  k )  4a( x  h)
k
where F (a, 0) the focus and (h,is)the vertex of
is
parabola.
× The parametric equation of a parabola:
x  at 2 ,
×
y  2at
The general equation of a parabola:
ax2  by 2  2 gx  2 fy  c  0
with either a=0 or b=0 but both not zero at the same
time.

19. 12.4 Equations of an Ellipse
An ellipse is a curve which is the locus of a variable
point which moves in a plane so that the sum of its
distance from two fixed points remains a constant.
The two fixed points are called foci.
P’(x,y)
P’’(x,y)

20. Let PF1+PF2 = 2a where a > 0
( x  c)  y  ( x  c)  y  2a
2
2
2
2
( x  c)  y  2a  ( x  c)  y
2
2
2
2
( x  c)  y  4a  4a ( x  c)  y  ( x  c)  y
2
2
2
2
2
4a ( x  c) 2  y 2  4cx  4a 2
a ( x  2 xc  c  y )  c x  2a cx  a
2
2
2
2
2
2
2
4
a 2 x 2  2a 2 xc  a 2c 2  a 2 y 2  c 2 x 2  2a 2cx  a 4
2

21. (a  c ) x  a y  a  a c
2
2
2
2
2
4
2 2
(a  c ) x  a y  a (a  c )
2
2
2
2
Let b  a  c
2
2
2
2
2
2
2
b x a y a b
2
2
2
2
2
2 2
2
x
y
 2 1
2
a
b
standard equation of
an ellipse

22. 12.4 Equations of an Ellipse
major axis = 2a
vertex
lactus rectum
minor axis = 2b
length of semi-major axis = a
length of the semi-minor axis = b
2b 2
length of lactus rectum =
a

23. 12.4 Equations of an Ellipse
AB
major axis
CD
minor axis
A, B, C and D
vertices
O
centre
PQ
focal chord
F
focus
RS, R’S’
latus rectum

24. 12.4 Equations of an Ellipse

25. 12.4 Equations of an Ellipse
Other form of Ellipse
2
2
x
y
 2 1
2
b
a
where a2 – b2 = c2
and a > b > 0

26. 12.4 Equations of an Ellipse
Furthermore,
x2 y2
(1) Given an ellipse 2  2  1, where a  b  0,
a
b
the length of the semi - major axis is a and that
of the semi - minor axis is b.
x2 y2
(2) Given an ellipse 2  2  1, where a  b  0,
b
a
then its foci lie on the y - axis, the length of the
semi - major axis is a and that of the semi - minor
axis is b.

27. 12.4 Equations of an Ellipse
( x  h) ( y  k )
(3) The equation

 1, represent an
2
2
a
b
ellipse whose centre is at (h, k ) and whose axes are
2
2
parallel to the coordinate axes.
y
( x  h) 2 ( y  k ) 2

1
2
2
a
b
(h, k)
O
x

28. DIFFERENT FORMS OF EQUATIONS OF ELLIPSE

The standard equation of ellipse:
x2 y 2
 2  1, a  b and c 2  a 2  b2
2
a b
where F (c, 0) the foci of the ellipse.
are
×
The parametric equation of an ellipse:
x  a cos  ,
y  b sin 

29. 12.7 Equations of a Hyperbola
A hyperbola is a curve which is the locus of a variable
point which moves in a plane so that the difference of
its distance from two points remains a constant. The
two fixed points are called foci.
P’(x,y)

30. Let |PF1-PF2| = 2a where a > 0
| ( x  c)  y  ( x  c)  y | 2a
2
2
2
2
( x  c)  y  2a  ( x  c)  y
2
2
2
2
( x  c)  y  4a  4a ( x  c)  y  ( x  c)  y
2
2
2
2
2
 4a ( x  c) 2  y 2  4cx  4a 2
a ( x  2 xc  c  y )  c x  2a cx  a
2
2
2
2
2
2
2
4
a 2 x 2  2a 2 xc  a 2c 2  a 2 y 2  c 2 x 2  2a 2cx  a 4
2

31. (c  a ) x  a y  a c  a
2
2
2
2
2
2 2
4
(c  a ) x  a y  a (c  a )
2
2
2
2
Let b  c  a
2
2
2
2
2
2
2
b x a y  a b
2
2
2
2
2
2 2
2
x
y
 2 1
2
a
b
standard equation of
a hyperbola

32. transverse axis
vertex
lactus rectum
conjugate axis
2b 2
length of lactus rectum =
a
length of the semi-transverse axis = a
length of the semi-conjugate axis = b

33. 12.7 Equations of a Hyperbola
A1, A2 vertices
A1A2
transverse axis
YY’
conjugate axis
O
centre
GH
focal chord
CD
lactus rectum

34. 12.7 Equations of a Hyperbola
asymptote
b
equation of asymptote : y   x
a

35. 12.7 Equations of a Hyperbola
Other form of Hyperbola :
2
2
y
x
 2 1
2
a
b

36. Rectangular Hyperbola
If b = a, then
2
2
x
y
 2 1
2
a
b
2
2
y
x
 2 1
2
a
b
x y a
2
2
2
y x a
2
2
2
The hyperbola is said to be rectangular hyperbola.

37. equation of asymptote :
x y 0

38. 12.7 Equations of a Hyperbola
Properties of a hyperbola :
( x - h) 2 ( y - k ) 2
(1) The equation

 1 represents a
2
2
a
b
hyperbola with centre at (h, k ), transverse axis
parallel to the x - axis.
( x - h) 2 ( y - k ) 2
(2) The equation 
 1 represents a
2
2
a
b
hyperbola with centre at (h, k ), transverse axis
parallel to the y - axis.

39. 12.7 Equations of a Hyperbola
Parametric form of a hyperbola :
 x  a sec 

 y  b tan 
where  is a parameter.
 the point (a sec  , b tan  ) lies on the
2
2
x
y
hyperbola 2  2  1.
a
b

40. 12.8 Asymptotes of a Hyperbola
2
2
x
y
The hyperbola 2  2  1, where a, b
a
b
are positive constants, has two asymptotes
x y
  0.
a b

41. 12.8 Asymptotes of a Hyperbola
Properties of asymptotes to a hyperbola :
x2 y2
(1) The hyperbola - 2  2  1 has two asymptotes
a
b
x y
   0.
a b
( x  h) 2 ( y  k ) 2
(2) The hyperbola

 1 has two
2
2
a
b
xh yk
asymptotes

 0.
a
b

42. 12.8 Asymptotes of a Hyperbola
Properties of asymptotes to a hyperbola :
( x  h) ( y  k )
(3) The hyperbola 

 1 has two
2
2
a
b
xh yk
asymptotes 

 0.
a
b
2
2

43. Simple Parametric Equations and Locus Problems
x = f(t)
y = g(t)
parametric equations
parameter
Combine the two parametric equations into
one equation which is independent of t.
Then sketch the locus of the equation.

44. Equation of Tangents to Conics
general equation of conics :
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
Steps :
dy
(1) Differentiate the implicit equation to find .
dx
dy
(2) Put the given contact point (x1, y1) into
dx
to find out the slope of tangent at that point.
(3) Find the equation of the tangent at that point.

45. THE GENERAL EQUATION OF SECOND DEGREE
Ax 2  By 2  Gx  Fy  C  0

Case I: IfA  B  0, the equation represents a circle with centre
G
F
G
(
,
at 2 A 2 A )
and radius 4 A  4FA  C
A
Case II: If A  Band both have the same sign, the equation
represents the standard equation of an ellipse in XY-coordinate
G
F
X  x
and Y  y 
system, where
2A
2B
Case III: If A  B and both have opposite signs, the equation
represents the standard equation of hyperbola in XY-coordinate
G
F
system, where X  x  2 A and Y  y  2( B)
Case IV: If A  0 or B  0 ,the equation represents the standard
equation of parabola in XY- coordinate system, where
2
2



2
2
G
C
G2
X  x
and Y  y  
2A
F 4 AF

46. THE DISCRIMINANT TEST
With the understanding that occasional
degenerate cases may arise, the quadratic
curve Ax2  Bxy  Cy 2  Dx  Ey  F  0 is
2
 a parabola, if B  4 AC  0
2
 an ellipse, if B  4 AC  0
2
 a hyperbola, if B  4 AC  0


47. CLASSIFYING CONIC SECTION BY ECCENTRICITY





In both ellipse and hyperbola, the eccentricity is the ratio of the
distance between the foci to the distance between the vertices.
Suppose the distance PF of a point P from a fixed point F (the
focus)is a constant multiple of its distance from a fixed line (the
directrix).i.e. PF  e.PD where e is the constant of
,
proportionality. Then the path traced by P is
(a). a parabola if e  1
(b). an ellipse of eccentricity e if e  1
(c). a hyperbola of eccentricity e if e  1

48. Conics
Parabola
Ellipse
Hyperbola
PF = PN
PF1 + PF2 = 2a
| PF1 - PF2 | = 2a
Graph
Definitio
n

49. Conics
Parabola
Ellipse
Hyperbola
x2 y2
 2 1
2
a
b
x2 y2
 2 1
2
a
b
Graph
Standard
Equation
y  4ax
2

50. Conics
Parabola
Ellipse
Hyperbola
x = -a
a
x  , e  PF1
e
PN
a
PF1
x  ,e
PN
e
Graph
Directrix

51. Conics
Parabola
Ellipse
Hyperbola
Graph
Vertices
(0,0)
A(-a,0), B(a,0),
C(0,b), D(0,-b)
A1(a,0), A2(-a,0)

52. Conics
Parabola
Ellipse
Hyperbola
major axis = AB
minor axis =CD
transverse axis =A1A2
conjugate axis =B1B2
where B1(0,b), B2(0,-b)
Graph
Axes
axis of
parabola = the
x-axis

53. Conics
Parabola
Ellipse
Hyperbola
4a
2b 2
a
2b 2
a
Graph
Length of
lantus
rectum LL`

54. Conics
Parabola
Ellipse
Hyperbola
----
----
b
y x
a
Graph
Asymptotes

55. Conics
Parabola
Ellipse
Hyperbola
Graph
Parametric
representation
of P
2
(at ,2at )
(a cos  , b sin  ) (a sec , b tan )