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