Mathematics Conic Section

Conic Section

Point, pair of straight lines, circle, parabola, ellipse and hyperbola are called conic section because they can be obtained when a cone (or double cone) is cut by a plane.

The mathematicians associated with the study of conics were Euclid, Aristarchus and Apollonius. Most of the objects around us and in space have shape of conic-sections. Hence study of these becomes a very important tool for present knowledge and further exploration.

Section of Right Circular Cone by Different Planes:

(1) When a double right circular cone is cut by a plane parallel to base at the common vertex, the cutting profile is a point.

(2) When a right circular cone is cut by any plane through its vertex, the cutting profile is a pair of straight lines through its vertex.

(3) When a right circular cone is cut by a plane parallel to its base the cutting profile is a circle.

(4) When a right circular cone is cut by a plane parallel to a generator of cone, the cutting profile is a parabola.

(5) When a right circular cone is cut by a plane which is neither parallel to any generator I axis nor parallel to base, the cutting profile is an ellipse.

(6) When a double right circular cone is cut by plane, parallel to its common axis, the cut profile is hyperbola

Hence a point, a pair of intersecting straight lines, circle, parabola, ellipse and hyperbola, all are conicsections. All the conic sections are plane or two dimensional curves.

The conic section is the locus of a point which moves such that the ratio of its distance from a fixed point(focus) to perpendicular distance from a fixed straight line (directrix) is always constant (e). Here `e` is called eccentricity of conic i.e.,

`(PS)/(PM)=e`

A line through focus and perpendicular to directrix is called- axis. The vertex of conic is that point where the curve intersects its axis.

`(PS)/(PM)=e=> PS^2=e^2PM^2`

`=> (x-alpha)^2+(y-beta)^2=e^2((Ax+By+C)/(sqrt(A^2+B^2)))^2`

Simplification shall lead to the equation of the form `ax^2 + by^2 + 2hxy+ 2gx + 2fy+ c = 0`

Distinguishing various conic :

The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also
upon the value of the eccentricity e. Two different cases ari se.
`text(Case-1 : When The Focus Lies On The Directrix (De-generated conic) :)`

In this case `Delta = abc+ 2fgh - af^2 - bg^2 - ch^2 = 0` `&` the general equation of a conic represents a pair
of straight lines if

`e > 1` i.e. `h_2 > ab` the lines will be real & distinct, intersecting at S.

`e = 1` i.e. `h_2 = ab` the lines will coincident.

`e < 1` i.e. `h_2 < ab` the lines will be imaginary.

`text(Case-II : When The Focus Does Not Lie on the Directrix (Non de-generated conic):)`
In this case `Delta = abc + 2fgh - af^2 - bg^2 - ch^2 ne 0` and conic represent

Definition of various terms related to a conic :

(I) Focus : The fixed point is called a focus of the conic.

(2) Directrix: The fixed line is called a directrix of the conic.

(3) Axis : The line passing through the focus and perpendicular to the directrix is called the axis of the conic.

(4) Vertex: The points of intersection of the conic and the axis are called vertices of the conic.

(5) Centre: The point which bisects every chord of the conic passing through it, is called the centre of the conic.

(6) Latus-rectum: The latus-rectum of a conic is the chord passing through the focus and perpendicular to the axis.

(7) Double ordinate : A chord which is perpendicular to the axis of parabola or parallel to its directrix.