Conic Sections

       
           

The Conic section is a part of analytic geometry, conic sections means that it is a curve obtained by the intersection of a cone along with it’s respective plane. It can also be defined as the second degree plane algebraic curve. In mathematical language the conic section can be defined as set of all those points which are at fixed distance ratio to some point and that point is called focus, line is called directrix and fixed ratio of distance is called eccentricity.
Generally we have three types of conic sections that are hyperbola, parabola and the ellipse, the circle is also considered as conic section but it is just a special case of ellipse.
Types of conic sections are given below:
Ellipse: Those conic sections with the eccentricity less than one (1) are considered as ellipse.
               Equation = [(x2/a2) + (y2/b2) = 1].
Parabola: Those conic sections with the eccentricity equal to one (1) are considered as parabola.
                Equation = [y2 = 4ac].
Hyperbola: Those conic sections with the eccentricity greater than one (1) are considered as hyperbola.
               Equation = [(x2/a2) - (y2/b2) = 1].
Circle can also be defined in focus - directrix definition and conic section with the eccentricity zero (0) is called circle. The circle and even ellipse are formed by the intersection of a cone and a plane. It is necessary in the case of circle that the cutting plane must be parallel to the plane of generating circle.
Equation = [x+ y= a2].
Latus Rectum: In conic section latus rectum can be defined as the line parallel to the directrix and passes through the focus.
Latus rectum is four times of the focal length in parabola.
The diameter in the circle is the latus rectum.
It is (2b2/a) in the case of ellipse.
General equation of conic section is:
P(x2) + Q(xy) + R(y2) + S(x) + T(y) + U = 0,
Here (P, Q, R….) are the factors.
As these are plane curves so we use x, y- coordinates. From this general equation we can form or create the equations of circle, parabola, ellipse and also even hyperbola.
 
 

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