Chapter:11conic

sections.

Mathematicsprojectwork.

NAME: MEHUL KUMAR DAS.

CLASS: XI, COMMERCE.ROLL.NO: 30

3RD UNIT TEST PROJECT.

SUBMITTED TO: V.S CHAUHAN SIR.

INTRODUCTION: This chapter is about parabola,

hyperbolas, circles, ellipses. the names parabola and

hyperbola are given by Apollonius.

These curves are in fact, known as conic sections or more commonly conics because they can be obtained as intersections of a plane with double napped right circular cones.

conics

Conics:

Sections of a cone: When β= 90, the section is a

circle. When α < β < 90, the section is

a ellipse. When β =α the section is a

parabola. When 0 ≤ β < α, the plane cuts

through both the nappes and the curves of intersection is a hyperbola.

Degenerated conic sections:

When the plane cuts at the vertex of the cone, we have the following different cases:

When α < β ≤ 90 then the section is a point.

When β = α the plane contains a generator of the cone and the section is a straight line.

When 0≤ β < α the section is a pair of intersecting straight lines.

Circle: A circle is the set of all the points in

a plane that are equidistant from a fixed point in the plane.

The fixed point is called the centre of the circle and the distance from the centre to a point on the circle is called the radius of the circle.

Given C(h,k) be the centre and the r the radius of the

circle.Let P(x,y) be any point on the circle Then by distance

formula, we have:

: (x - h)2 + (y - k)2 = r2

Parabola: A parabola is the set of all the

points in a plane that are equidistant from a fixed line and a fixed point in the plane.

The fixed line is called the directrix of the parabola and the fixed point F is called the focus.

A line through the focus and perpendicular to the directrix is called the axis of the parabola.

The point of intersection of parabola with the axis is called the vertex of the parabola.

parabola

Standard equations of parabola:

Left handed parabola: y²= 4ax. X=-a, f(a.0).

.

Right handed parabola: y²=-4ax. X=+a, f(-a,0).

Continue: Upward parabola: x²=4ay. Y=-a, f(0,a).

Downward parabola: x²=-4ay. Y=a, f(0.-a).

Latus rectum of a parabola: Latus rectum of a parabola is a line

segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola

Length of latus rectum= 4a.

Ellipse: An ellipse is the set of all the points in a

plane, the sum of whose distances from two fixed points in the plane is a constant.

Continues: Major axis= 2a. Minor axis=2f Foci=2c.

Relationship: A²=b²+c². C=√a²-b².

Eccentricity: The eccentricity of an

ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.It is denoted by e= c⁄a.

Standard equations of an ellipse.

Latus rectum of an ellipse: Latus rectum of an ellipse is

a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.

Length of the latus rectum of an ellipse:

Hyperbola: A hyperbola is the set of all the

points in a plane, the difference of whose distances from two fixed points in the plane is a constant.

Eccentricity of hyperbola: Just like an ellipse, the ratio e=c/a, is called the eccentricity of the

hyperbola.

Standard equations of the hyperbola:

Latus rectum of hyperbola: Latus rectum of an

hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

Length of latus rectum in hyperbola:

2b2/a

Thank you.