PARABOLA VS HYPERBOLA

1. TOPIC NAME : PARABOLA VS HYPERBOLA

2. INTRODUCTION
 Hyperbola vs Parabola is an interesting difference to study in geometry.
Let us first see a few lines about these two curves and then go deep into
hyperbola vs parabola.
 Hyperbola is like an open curve with two branches. It is an intersection
of a plane with both halves of a double cone. The plane doesn’t need to
be parallel to the cone’s axis; the hyperbola will be symmetrical in any
case.
 A parabola is a curve, where any point is at an equal distance from a
fixed point (the focus), and a fixed straight line (the directrix).

3. HISTORY
 Menaechmus discovered Hyperbolae during his investigations of the
problem of doubling the cube but later called sections of obtuse
cones. The term hyperbola is believed to be coined by Apollonius of
Perga in his definitive work, the Conics, on the conic sections.
 Archimedes computed the area enclosed by a parabola and a line
segment by the method of exhaustion, 3rd century BC.
 Archimedes (Source)The name “parabola” by Apollonius, who
discovered many properties of conic sections. It means “application,”
referring to “application of areas”.

4. WHAT IS PARABOLA?
 Parabola is a plane curve which is mirror-symmetrical and
is approximately U-shaped.
 A plane curve generated by moving points so that its
distance from a fixed point is equal to its distance from a
fixed-line: the intersection of a right circular cone with a
parallel plane to an element of the cone.

5. GENERAL EQUATION OF PARABOLA
 General equation of parabola is given by,
 Y2=±4ax (directrix is parallel to y-axis),
&
 X2=±4by (directrix is parallel to x-axis)

6. WHAT IS HYPARBOLA?
 The pair of hyperbolas are formed by the
intersection of a plane and two equal right
circular cones on opposite sides of the same
vertex.
 A hyperbola is the locus of all those points in a
plane, such that the difference in the distances
from two fixed points in the plane is a
constant.

7. GENERAL EQUATION OF HYPARBOLA
 Taken as known the focus (h, k),
 The formula for the Hyperbola can be given by,
(x−h)2/a2–(y−k)2/b2=1,
 Here, a is the transverse axis & b is the conjugate axis.
Let, the directrix of parabola be y=mx+b, then the equation
is given as
[y–mx–b]2/[m2+1]= (x–h)2 + (y–k)2

8. ECCENTRICITY
 The eccentricity is the distance ratio from the
centre to a vertex and from the centre to a
focus(foci). It is denoted by e.
 Eccentricity of Hyperbola = ca
 Eccentricity of hyperbola is always ≥1.
 Since, c≥a
 The eccentricity of Parabola is 1.

9. LATUS RECTUM
 The line segments perpendicular to the
transverse axis through any of the foci such
that their endpoints lie on the curve are
defined as the latus rectum.
 The Latus Rectum in Hyperbola is given by
2b2/a
 The Latus Rectum of Parabola is given by 4a.

10. APPLICATIONS
 The main application of parabolas is their
reflective properties (lines parallel to the axis
of symmetry reflect the focus).
 Hyperbola is handy in real-world applications
like telescopes, headlights, flashlights,
lampshades, Gear transmission, structure for
Coal-fired Power Plants, etc.

11. THANK YOU