What is Conic Section?
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Conic section is a curve obtained by the intersection of the surface of a cone with a plane.
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In Analytical Geometry, a conic is defined as a plane algebraic curve of degree 2. That is, it consists of a set of points which satisfy a quadratic equation in two variables. This quadratic equation may be written in matrix form. By this, some geometric properties can be studied as algebraic conditions.
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Thus, by cutting and taking different slices(planes) at different angles to the edge of a cone, we can create a circle, an ellipse, a parabola, or a hyperbola, as given below
Source: ellipsesconicsections.weebly.com
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The circle is a type of ellipse, the other sections are non-circular. So, the circle is of fourth type.
Focus, Directrix and Eccentricity
The curve is also defined by using a point(focus) and a straight line (Directrix).
If we measure and let
a – the perpendicular distance from the focus to a point P on the curve,
and b – the distance from the directrix to the point P,
then a: b will always be constant.
\(a: b < 1\) for ellipse
\(a: b= 1\) for parabola as \(a= b\)
and \(a: b> 1\) for hyperbola.
Source: .wikimedia.org
Eccentricity: The above ratio a: b is the eccentricity.
Thus, any conic section has all the points on it such that the distance between the points to the focus is equal to the eccentricity times that of the directrix.
Thus, if eccentricity \(<1\), it is an ellipse.
if eccentricity \(=1\), it is a parabola.
and if eccentricity \(=1\), it is a hyperbola.
Source: https://3.bp.blogspot.com
For a circle, eccentricity is zero. With higher eccentricity, the conic is less curved.
Latus Rectum
The line parallel to the directrix and passing through the focus is Latus Rectum.
Source: i2.wp.com/c1.staticflickr.com
Length of Latus Rectum = 4 times the focal length
Length \(=\frac{2b^2}{a}\) where \(a =\frac{1}{2}\) the major diameter
and \(b =\frac{1}{2}\) the minor diameter.
= the diameter of the circle.
Ellipse has a focus and directrix on each side i.e., a pair of them.
Equations
General equation for all conics is with cartesian coordinates x and y and has \(x^2\) and \(y^2\) as
the section is curved. Further, x, y, x y and factors for these and a constant is involved.
Thus, the general equation for a conic is
\[Ax^2 + B x y + C y^2+ D x + E y + F = 0\]
Using this equation, following equations are obtained:
For ellipse, \(x^2a^2+y^2b^2=1\)
For hyperbola, \(x^2a^2-y^2b^2=1\)
For circle, \(x^2a^2+y^2a^2=1\) (as radius is a)
