In mathematics, a conic section(s) are non-degenerate curve(s) that are formed by the intersection of the surface of an infinite cone and and infinite plane. Hyperbola, ellipse, and parabola are together known as conic sections, or just conics, so called because they are the sections created by the intersection of a right circular cone and a plane.
Section of a Right Circular Cone by a Plane at Different Angles
Section I Section of a right circular cone by a plane which is passing through its vertex is a pair of straight lines. The pair of lines always passes through the vertex of the cone. In figure, the VA and VB are the pair of lines passing through the vertex of the right circular cone cut by a plane, in such a way that the plane passes through the vertex of the cone. Such type of conic is called a Degenerate Conic.
Section II Section of a right circular cone by a plane parallel to the base or inclined at an angle of 900 with the axis of the cone is a circle as shown in figure.
Section III Section of a right circular cone by a plane which is not parallel to any generator and not parallel or perpendicular to the axis of the cone is an ellipse.
The most well-knlwn members of this family of two-diamensional curves are the circle and the ellipse. These arise when the intersection is a closed curve, the circle is a special case of the ellipse in which the plane is exactly perpendicular to the axis of the cone.
Section IV Section of a right circular cone by a plane which is parallel to a generator line of the cone, is called a parabola.
Section V Family, if the intersection (of the plane with the cone)is an open curve, and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (see figure).
The points of intersection of the conic section and the axis is called the vertex of the conic section.
Any chord passing through the focus is called the focal chord of the conic section.
A straight line drawn perpendicular to the axis and terminated at both end the of the curve is a double ordinate of the conic section.
The double ordinate passing through the focus is called the latus rectum of the conic section.
The point which bisect every chord of the conic passing through it, is called the centre of kthe conic section.
Note: The above definitions are defined with the help of the parabola as shown in figure.
Recognisition of Conics
In the Cartesian Coordinate System, the graph of a quadratic equation in two variables is always a conic section. If the equation is of the form
then the discriminant of the above quadratic equation is
If then the Conic is a degenerate conic or the equation of conic represents an equation of pair of straight line.
If the equation represents a parallel pair of straight lines.
If the equation represents an intersecting pair of straight lines.
If the equation represents a point.
If then the Conic is a non-degenerate conic
If this equation represents a parabola.
If this equation represents an ellipse or an empty set.
If a = b and h = 0, it represents a circle.
If this equation represents a hyperbola.
If a + b = 0 and it represents a rectangular hyperbola.
Illustration 2: What type of conic does the equation xy = 1 represent?
Solution: Step I Comparing the equation with we have
and c = –1
Step II Calculating the value of the determinant to find whether to conic is a degenerate of non-degenerate conic.
Since, the discriminant Δ > 0 therefore the conic represents a non-degenerate conic.
Step III Also
Therefore, from STEP II and STEP III, the conic represented by the two degree equation is a non-degenerate conic and since the non-degenerate conic represents a hyperbola.
Centre of Conic
Centre of the conic is the point which bisect every chord of the conic passing through it. If is the equation of the conic section and if C is its centre, then the centre of the conic is
Shortcut Method to Find the Centre of Conic
Let is the equation of the conic section, then partially differentiate the equation w.r.t. x and then w.r.t. y. Such that
Patially differentiating w.r.t. x keepint y as Partially differentiating w.r.t. y keeping x as constant
constat, we have we have
⇒ ax+hy+g = 0 ........(1) ⇒ hx+by+f = 0 .........(2)
Solving equations (1) and (2),
which is the centre of the conic section.
Illustration 3: Find the centre of the conic
Solution: Let the equation of the conic is
Partially differentiating w.r.t. Partially differentiating w.r.t
x keeping y as constant, y keeping x as constant.
we have we have
2x-3y +4 = 0 .......(1) 3x+6y-22 = 0 .......(2)
Solving equation (1) and (2), we have C(x, y) =C(2,8/3)
A parabola is the locus of a point which moves in a plane such that its distance from the fixed point i.e. focus is always equal to its distance from a fixed line i.e. directrix.
where, e called eccentricity or contracity and is equal to the ratio of distances of any point
For a parabola, e = 1
Now, let S be the focus represented by the coordinates (a, 0) and the line passing through Z and M represents the directrix as can be seen in the figure.
Since, for a parabola, in general, we have
[by definition] ......(1)
If P approaches A, then M will approach to Z, such that
equation of directrix passing through Z and M is
which is a line parallel to the y-axis. Also, x-axis passes through the focus and is perpendicular to the directrix, hence, x-axis acts as the Axis of the conic section, parabola in this case. Such that the point A will be the vertex of the parabola