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Hyperbola Equation

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					                       Hyperbola Equation
Hyperbola Equation

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a
plane, which can be defined either by its geometric properties or by the kinds of
equations for which it is the solution set.

A hyperbola has two pieces, called connected components or branches, which are
mirror images of each other and resembling two infinite bows.

The hyperbola is one of the four kinds of conic section, formed by the intersection
of a plane and a cone. The other conic sections are the parabola, the ellipse, and
the circle (the circle is a special case of the ellipse).

Which conic section is formed depends on the angle the plane makes with the axis
of the cone, compared with the angle a line on the surface of the cone makes with
the axis of the cone.

If the angle between the plane and the axis is less than the angle between the line
on the cone and the axis, or if the plane is parallel to the axis, then the conic is a
hyperbola
                                              Know More About :- Intro to Precalculus


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Hyperbola in daily life can be seen in the form of a decoration or design. It is a
smooth curve and an unbounded case of the conic section, formed by the
Intersection of a plane with both halves of a double cone.

The shape of a Hyperbola is well-defined by its eccentricity e, which is a
dimensionless number always greater than one. A hyperbola can also be defined
as the locus of all points of the plane whose distances to two fixed points are a
constant.

Some important terms regarding hyperbola are:

Focus of Hyperbola: the two points on the transverse axis that are responsible for
the entire shape of the hyperbola.

Transverse Axis: the axis on which the two foci lie

The standard hyperbola equation is almost identical to the ellipse. If the
transverse axis of hyperbola is aligned with the x-axis of a Cartesian coordinate
system and is centered at (h, k), the general equation of hyperbola can be written
as (x - h)2/A2 - (y - k)2/B2=1

And if the alignment is along the y-axis the equation is given as
(y - h)2 / A2 - (x - k)2/B2=1 or (x - h)2/A2 - (y - k)2 / B2= - 1
Where h, k, a and b are real and positive Numbers

Example: Show that the line 4p – 3q = 9 touches the hyperbola 4p2 – 9q2 = 27.
 Solution: For the line y = mx + c to touch the hyperbola x2 / a2 – y2 / b2 = 1, the
condition is c2= a2m2 – b2

Here the hyperbola is given as: p2 / (27 / 4) – q2 / (27 / 9) = 1,

                                                  Learn More :- Dot Products Of Vectors


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i.e. here a2 = 27/4, b2 = 27/9 = 3,

And comparing the equation of the given line 4p – 3q = 9 with y = mx + c, we get
Slope, m = 4/3 and
        c = –3

∴ a2m2 – b2 = (27 / 4)(4 / 3)3 – 3 = 12 – 3 = 9 = (–3)2

or a2m2 – b2 = c2, thus satisfying the above condition

Hence the given line touches the given hyperbola.




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