CONIC SECTION

MATH-002Dr. Farhana Shaheen

CONIC SECTION

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a directrix.

CONICS

The three conic sections that are created when a double cone is intersected with a plane.

1) Parabola 2) Circle and ellipse 3) Hyperbola

CIRCLES

A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.

PARABOLA

PARABOLA: LOCUS OF ALL POINTS WHOSE DISTANCE FROM A FIXED POINT IS EQUAL TO THE DISTANCE FROM A FIXED LINE. THE FIXED POINT IS CALLED FOCUS AND THE FIXED LINE IS CALLED A DIRECTRIX.

P(x,y)

EQUATION OF PARABOLA

Axis of Parabola: x-axis Vertex: V(0,0) Focus: F(p,0) Directrix: x=-p

pxy 42

DRAW THE PARABOLA xy 62

pxy 42

PARABOLAS WITH DIFFERENT VALUES OF P

EQUATION OF THE GIVEN PARABOLA?

PARABOLAS IN NATURE

PARABOLAS IN LIFE

ELLIPSE: LOCUS OF ALL POINTS WHOSE SUM OF DISTANCE FROM TWO FIXED POINTS IS CONSTANT. THE TWO FIXED POINTS ARE CALLED FOCI.

ELLIPSE

a > b Major axis: Minor axis: Foci: Vertices: Center: Length of major axis: Length of minor axis: Relation between a, b, c

EQUATION OF THE GIVEN ELLIPSE?

EQUATION OF THE GIVEN ELLIPSE IS

EARTH MOVES AROUND THE SUN ELLIPTICALLY

DRAW THE ELLIPSE WITH CENTER AT(H,K)

ECCENTRICITY

ECCENTRICITY IN CONIC SECTIONS

Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity.

CIRCLE AS ELLIPSE

A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected by a plane perpendicular to the axis of the cone.

HYPERBOLA

HYPERBOLA

Transverse axis: Conjugate axis: Foci: Vertices: Center: Relation between a, b, c

HYPERBOLA WITH VERTICAL TRANSVERSE AXIS

ECCENTRICITY E = C/A

e = c/a e= 1 Parabola e=0 Circle e>1 Hyperbola e<1 Ellipse

ECCENTRICITY E ELLIPSE (E=1/2), PARABOLA (E=1) AND HYPERBOLA (E=2) WITH FIXED FOCUS F AND DIRECTRIX

HYPERBOLA

THANK YOU