Hsslive-Ch 11. Conic Section
Hsslive-Ch 11. Conic Section
Hsslive-Ch 11. Conic Section
CHAPTER 11
CONIC SECTION
1. Locus of a point: When a point moves in such a way that its path traced out is
called its locus. E.g.: sector of a circle, circle, conic, etc.
2. A conic is a curve obtained by slicing a cone with a plane which does not pass
through the vertex. It is also known as conic section.
3. Types of conics:
axis generator
ellipse
Upper nappe
vertex
hyperbola circle
Lower nappe
generator
parabola
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
4. Conic section as a locus: The conic section is the locus of a point which moves in a
plane so that its distance from a fixed point bears a constant ratio to its distance
from a fixed line.
5. The fixed point is called focus (S) and the fixed line is called directrix and the
constant ratio is called eccentricity of the conic, which is denoted by e.
6. If e=1 the conic is known as a parabola, if e 1 , the conic is known as an ellipse
and if e 1 , the conic is known as a hyperbola and if e 0 , the conic is known as a
circle.
PARABOLA
Standard Equation
Four types of parabolas:
Graph
Vertex A 0, 0 A 0, 0 A 0, 0 A 0, 0
Focus S a, 0 S a, 0 S 0, a S 0, a
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
directrix
Axis x axis x axis y axis y axis
ELLIPSE
Definition2: An ellipse is the set of all points in the plane whose distances from a fixed
point in the plane bears a constant ratio, less than one, to their distances
from a fixed line in the plane.
Definition3: An ellipse the set of all points in a plane, the sum of whose distances from
two fixed points in the plane is a constant.
Definition4: If e<1, then the conic is known as an ellipse.
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
PROPERTIES
Vertices A a, 0 , A a, 0 A 0, a , A 0, a
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
rectum
Length of major axis 2a 2a
Length of minor axis 2b 2b
Note:
1. b 2 a 2 1 e 2
2. Centre of the ellipse is the point of intersection of the major and minor axis.
3. Foci are the point of intersection of major axis and the latus recta.
4. 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 is known as eccentricity.
c
The eccentricity, e , where c a 2 b 2
a
HYPERBOLA
Definition2: A hyperbola is the set of all points in the plane whose distance from a
fixed point in the plane bears a constant ratio, greater than one, to their
distances from a fixed line in the plane.
Definition3: A hyperbola the set of all points in a plane, the sum of whose distances
from two fixed points in the plane is a constant.
Definition4: If e>1, then the conic is known as a hyperbola.
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
PROPERTIES
Vertices A a, 0 , A a, 0 A 0, a , A 0, a
HSSLIVE.IN rchciit@gmail.com |
Formula Master Part I
Note:
1.
b2 a 2 e2 1
2. Centre of the hyperbola is the point of intersection of the transverse axis and the
conjugate axis.
3. Foci are the point of intersection of transverse axis and the latus recta.
4. The ratio of the distances from the centre of the hyperbola to one of the foci and to
one of the vertices of
5. the hyperbola is known as eccentricity.
c
The eccentricity, e , where c a 2 b 2
6. a
HSSLIVE.IN rchciit@gmail.com |