Conic SectionsConic SectionsConic section can be defined as a curve which is made by the intersection of a
cone that resides on a plane. And in other terms it can be assumed as the plane
algebraic curve with the degree of two.The general equation of any conic section
is given by:
Sp2 + Tpq + Uq2 + Vp + Wq + X = 0;
If the value of T = 0 then we will see the `S' and `U' in the equations. Name of
conic section Relationship of A and C. Parabola S = 0 or U = 0 but both the values
of `S' and `U' are never equals to 0. Circle - In case of circle the value of `S' and `U'
are both equal.
Ellipse - In case of ellipse the sign of `S' and `U' are same but `S' and `U' are not
equal. Hyperbola - In case of hyperbola the signs of `S' and `U' are opposite.
Let's have small introduction about all conic sections. Hyperbola can be defined as
a line in a graph that has curve shape.
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Generally the equation of hyperbola is given by: x2 / F2 - y2 / G2= 1; this is
equation of hyperbola. In geometry a parabola is a special type of curve that has
its own shape just like an arc and the point situated on a parabola is always
equidistance from the locus and the directrix.
The general form of parabola is given as: (s + t)2 + s + t + = 0; This
equation is obtained from the general conic sections equation. The equation is
given by: Sp2+ Tpq + Uq2 + Vp + Wq + X = 0; And the equations for a general
form of parabola with the focus point F(s, t) and a directrix in the form: pa + qy
+ c = 0; This is the equation of parabola. We can determine the conic section into
three types: The three types of conic section are:
1. Parabola - Parabola has eccentricity is equal to 1.
2. Circle and Ellipse - Its eccentricity is larger than 0 smaller than 1.
3. Hyperbola - It has eccentricity value greater than 1. These all are three types
of conic sections and its eccentricity.
There are some special categories defined for a circle that is - in some of the
cases it is assumed as fourth type conic section and in some cases it is assumed
as a special type of ellipse.
If we intersect a plane and a circular then we get conic section. Some equation of
conic section is defined which are shown below:
The general equation of a circle is given by: - x2 + y2 = a2. The equation of an
ellipse is given by: - x2 / a2 + y2 / b2 = 1.
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The general equation of a parabola is given by: - y2 = 4ax. And the general
equation of hyperbola is given by: - x2 / a2 - y2 / b2 = 1.
We can write all of the above equations in standard form and in parametric form.
The main and important area of conic section where it is used, they are: -
astronomy and projective geometry.
When the Intersection of Right Circular Cone and a plane which is also parallel to
an element of the cone and the locus points of cone are equidistant from a fixed
Point then we get a Plane Curve or a Parabola.
The general form is:
(p + q)2 + p + q + = 0;
This above equation is obtained from the general conic sections equation which is
given below:
Ap2+ Bpq + Cq2 + Dp + Eq + F = 0;
And the equations for a general form of parabola with the focus point F(s, t) and a
directrix in the form:
pa + qy + c = 0;
This is the equation of parabola.
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