Post on 15-Jul-2015
CONIC SECTIONS
XI C
α β
THE INTERSECTION OF A PLANE WITH A CONE, THE SECTION SO OBTAINED IS CALLED A
CONIC SECTION
V
m
Lower nappe
Upper nappe
Axis
Generator
l
This is a conic section.
TYPES OF CONIC SECTIONS
CIRCLE
A CIRCLE IS THE SET OF ALL
POINTS ON A PLANE THAT ARE EQUIDISTANT FROM A FIXED
POINT ON A PLANE.
O
P (x,y)
(h,k)C
P(x,y)
O (0,0)
x² + y² = r² (x – h) ² + (y – k) ² = r²
α β
When β = 90°, the section is a circle
Standard Equation General Equation
TYPES OF CONIC SECTIONS
ELLIPSEAN ELLIPSE IS THE SET
OF ALL THE POINTS ON A PLANE,
WHOSE SUM OF DISTANCES FROM TWO FIXED TWO REMAINS
CONSTANT.
PP P
F F
¹
³²
²¹
α β
O
(0,c)
(0,-c)
(-b,0) (b,0)
(0,-a)
(0,a)
x² y²
a² b²— —+ = 1—+
x² y²
b² a²— = 1
(-c ,0) (c, 0)
When α < β < 90°, the section is an ellipse
Vertical Ellipse
Horizontal Ellipse
(0,-b)
(0,b)
(a,0)(-a,0)
.
TYPES OF CONIC SECTIONS
A PARABOLA IS THE SET OF ALL POINTS IN A PLANE THAT
ARE EQUIDISTANT FROM A FIXED POINT
A
B
V
PARABOLA
(VERTEX)
F ( focus)
1 2 3 4O
P
1
P2
α
β
F(a,0)O
x =
-a
y² = 4ax
X' X
Y'
Y
F(-a,0) O
x =
+a
y² = -4ax
X' X
Y'
Y
F(0,-a)
O
y = a
x² = 4ay
X' X
Y'
Y
F(0,a)
O
y = -a
x² = -4ay
X' X
Y'
Y
When α = β, the section is an parabola
Horizontal Parabola Horizontal Parabola
Vertical Parabola Vertical Parabola
TYPES OF CONIC SECTIONS
HYPERBOLA
F ( focus)V
(vertex)
A
B
A HYPERBOLA IS THE SET OF ALL POINTS,THE DIFFERENCE OF WHOSE DISTANCES FROM TWO
FIXED POINTS IS CONSTANT
V(vertex)
F ( focus)
α β
Transverse axis
F
Conjugate axis
F(c ,0)(a ,0)( -c ,0)(-a ,0)
O
F
F (0 ,c)
(0 ,a)
(0 ,-c)
(0 ,-a)O
¹
¹
²
²
x² y²
a² b²— —- = 1
-
y² x²
a² b²— —- = 1
When 0 ≤ β < α; the plane cuts through both the nappes & the curves of intersection is a hyperbola
HYPERBOLIC PARABOLOIDSUNDIAL
THERMAL POWER PLANT
Conic Section Standard Eq. General Eq.
Circle x² + y² = r² (x – h) ² + (y – k) ² = r²
Parabola y² = 4ax (y-k)² = 4a(x+h)
Ellipse
Hyperbola
x² y²
a² b²— —+ = 1
(x-h)² (y-k)²
a² b²— + — = 1
x² y²
a² b²— —- = 1
(x-h)² (y-k)²
a² b²— - — = 1