By H.K.MEENA
PGT (Maths)
KV BEAWAR (Raj)

In mathematics, a conic section (or just conic ) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.
.
Apollonius

Sections of a cone
Let l be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle α
We rotate the line m around the line l in such a way that the angle
α remain constant. Then the surface generated is a doublenapped right circular hollow cone
Upper nappe herein after referred as cone
If we take the intersection of a nappe plane with a cone ,the section so
Lower obtained is called a conic section.
Thus ,conic sections are the curves obtained by intersecting a right circular cone by a plane.
Axis
α
V m
Generator l

Sections of a cone
When the plane cuts the nappe(other than the vertex) of the cone, we have the following situations:
When

=90 0 , the section is a circle
When
α 
90 0 ,section is an ellipse
When

=α ,the section is parabola
(in each of the 3 situations, the plane cuts entirely across one nappe of the cone)
When 0≤
 α ,the plane cuts through both the nappes and the curves of intersection is a hyperbola
α


When

=90 0 , the section is a circle
α

α 
When
α 
90 0 ,section is an ellipse

When

=α ,the section is parabola
α

When
0≤  α
,the plane cuts through both the nappes and the curves of intersection is a hyperbola

Degenerated conic sections
If α
 ≤90 0 then the section is a point
α

If

=α, section is a line
If 0≤  α, section is pair of straight line

α
α


Two well-known conics are the circle and the ellipse . These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone.
If the plane is parallel to a generator line of the cone, the conic is called a parabola
Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola .
(In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.)
The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line, or a pair of intersecting lines, are often excluded from the list of conic sections.

Circle
Ellipse (h) Parabola (h)
Hyperbola (h)
Definition:
A conic section is the intersection of a plane and a cone.
Ellipse (v)
Parabola (v)
Hyperbola (v)

Conics as point loci
Each type of conic can be defined as the locus of all points P with a certain property :
•
Circle: dist(P,C) = r, where C is a fixed point (the center), and r is a fixed constant (the radius).
•
Parabola: dist(P,F) = dist(P,L), where F is a fixed point (the
focus), and L is a fixed line (the directrix) not containing F.
•
Ellipse: dist(P,A) + dist(P,B) = d, where A,B are distinct fixed points (the foci), and d > dist(A,B) is a fixed constant (the major
diameter)
•
Hyperbola: dist(P,A) − dist(P,B) = d, where A,B are distinct fixed points (the foci), and d < dist(A,B) is a fixed constant.
In projective geometry, each of the conics can be defined as the locus of all points equidistant from a fixed point (the focus) and a fixed curve (the directrix); All conics have identical properties in projective space, and every theorem in projective geometry true of one conic section is true of all the others.

Eccentricity
Ellipse (e=1/2) , parabola (e=1) and hyperbola (e=2) with fixed focus F and directrix.
The four defining conditions above can be combined into one condition that depends on a fixed point F (the focus), a line L (the
directrix) not containing F and a nonnegative real number e ( the eccentricity ). The corresponding conic section consists of all points whose distance to F equals e times their distance to L.
For 0 < e < 1 we obtain an ellipse , a hyperbola .
for e = 1 a parabola , and for e > 1

Semi-latus rectum and polar coordinates
Semi-latus rectum in the case of an ellipse
The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to the semi-major axis a, and the semi-minor axis
b, by the formula al = b 2 , or l = a(1 − e 2 ).

Conic sections are important in astronomy : the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.
In projective geometry , the conic sections in the projective plane are equivalent to each other up to projective transformations.

Conics in analytic geometry
In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form then:
•if h 2 < ab , the equation represents an ellipse;
•if we also have a = b and h = 0, the equation represents a circle;
•if h 2 = ab , the equation represents a parabola;
•if h 2 > ab , the equation represents a hyperbola;
•if we also have
a + b = 0 , the equation represents a rectangular hyperbola.
Note that these a and b are just polynomial coefficients, not the lengths of semi-major/minor axis as defined in the previous sections.

Conic Sections
Intersections of parallel planes and a double cone, forming ellipses, parabolas, and hyperbolas respectively .

Description
Hyperbola, ellipse, and parabola are together known as conic sections, or just conics. So called because they are the intersection of a right circular cone and a plane.
Conics can be defined as follows. Given a line d and a point F not on d, conics is the locus of points P such that: distance[P,F]/distance[P,d] == e, where e is a given constant. F is called the focus of the conic, d the directrix, and e the eccentricity.
If 0 < e < 1, the conics is an ellipse.
If e == 1, it is a parabola, if e > 1, it is a hyperbola and has two branches.
below: figure shows conic sections with eccentricity
{.2, .4, .6, .8, 1, 1.5, 2, 2.5, 3}.
Their focus is at the origin.
The directrix is the white line x==1.

The directrix of a conic section is the line which toghether with the point known as the focus,serves to define a conic section as th locus of point whose distance being the constant of proportionality. If the ratio r =1, the conic is a parabola, if r < 1 it is an ellipse, and if r > 1, it is hyperbola.
Hyperbola and non circular ellipses have two distinct foci and two associated directrices each directrix being perpendicular to the line joining the two foci

The equations for each of the curves types (centered at the origin) are given on the right.
Note that the equations for the circle and ellipse are not dependent on the axis, the parabola and hyperbola come in two variants depending on whether they are aligned vertically or horizontally

Nappes
If the cone is imagined to extend until it goes to a point and then enlarges again then the two cones are known as "nappes". A plane that intersects just one nappe is either an ellipse or parabola, if it intersects both nappes then it is a hyperbola.
Focus
The solid dots on the right indicate the focus of each curve. For the ellipse the focus distance is a 2 - b 2 and for the hyperbola it is a 2 + b 2 .

Solve the following:-
Y
45
A
Q1. The focus of the parabolic mirror is at a distance of 5 cm from its vertex. If the mirror is
O
X
45 cm deep, find the distance AB
B
Q2. An arch is in the form of a semi-ellipse . It is 8m wide and 2m high at the centre. Find the height of the arch at a point 1.5m from one end.
Q3. An equilateral triangle is inscribed in the parabola y 2 =4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.