K.J. SOMAIYA COMPREHENSIVE COLLEGE OF EDUCATION TRAINING AND RESEARCH

NAME:- S.LAKSHMIR.NO:- 55

TOPIC:- CONIC SECTION (PARABOLA)

CONTENT

• HISTORY• DEFINITION• IMPORTANT TERMS• STANDARD EQUATION OF PARABOLA• DERIVATION• SKETCHING OF A PARABOLA• EVALUATION• SOLUTION

HISTORY

• Discovered by the Greek mathematician Manaechmus.

• He was the tutor of the great Alexander.• The theory applies to the

– Lenses of Telescopes.– Microscopes.– Weather prediction.– Communication by Satellites.– Construction of buildings and bridges.

DEFINITION

• A conic section is the locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line.

Note:- Fixed Point -> Focus

Fixed Line -> Directrix

Constant Ratio -> Eccentricity ( e )

P ( MOVING POINT )

S ( FIXED POINT )

M

L

K

D

I

R

E

C

T

R

I

X

Meaning of conic section through diagram

IMPORTANT TERM

1) AXIS:-The straight line passing through the focus and perpendicular to the directrix is called Axis.

M

B

A

S

P

IMPORTANT TERM

2) VERTEX :- A point of intersection of a conic with its axis is called a Vertex.

M

A

P1

B

S

P2

P3

STANDARD EQUATION OF A PARABOLA

y2=4ax

S (a,0)

P (x,y)

A (0,0) Z(-a,0) X

Y

T

DERIVATION

focus- S(a,0) vertex A(0,0) directrix LMAX - +ve Axis AY- +ve AxisMoving point P(x,y)By definition PS = PM √(x-a)2+(y-0)2 = x+a (x-a)2 +y2 = (x+a)2 x2 + a2 -2ax + y2 = x2 + a2 + 2ax y2 = 4ax

SKETCHING OF PARABOLA

-5a

-4a

-3a

-2a

5a4a 3a2a a-a

-a-2a-3a-4a-5a

a

2a

3a

4a

5a

X 0 a a 4a 4a

Y 0 2a -2a -4a 4a

EVALUATION

Q1. Define the following terms.i) Conic section ii) Axisiii) VertexQ2. Tick the correct answer.i) Who discovered Conic Section?

a) Manaechmus

b) AlexanderANS - a

EVALUATION

ii) The fixed point is known as

a) Vertex

b) Focus

ANS - b