In math, conic section is the section or part of the cone. It is simply known as the conics. Imagine when the plane intersecting cone precisely, then a curve is obtained which is called as Conic section or simply a conics. These curves are different according to the intersection of the cone with the plane. These are circle, parabola, hyperbola, and ellipse. These are the following different types of the conics-
(i) Circle:- A circle may be defined as the set of all points in the plane that are equidistant from a fixed point in the plane.
This fixed point is said to be the centre of the circle and the distance between the centre and any point on the circle is called the radius of the circle as shown below in figure (a).
The equation of the circle can be derived with a given centre and radius as in figure (b).
Let C(h, k) be the centre and r be the radius of the circle. If P(x, y) be any point on the circle [figure- (b)]. Then, by the definition of the circle,
CP = r , by the distance formula, we have
$\sqrt{(x-h)^2+ (y-k)^2}$ = r
i.e.
(x-h)2 + (y-k)2 = r2
This is the equation of the circle with centre at (h, k) and radius r.
When a cone is intersected precisely then the obtained part is known as a curve or conic section. This theory was given by the Apollonius. These curves can be obtained as intersections of a plane with a double napped right circular cone. These curves have very wide range of applications in fields such as planetary motion, design of telescopes and antennas, reflectors in flashlights and automobile headlights, etc.
Cross Section of a Cone
Let "l" be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at an angle ?, as shown in the following figure-
If line l is rotated such that the angle ? remains constant. Then a double-napped surface is generated right circular hollow cone herein cone herein after referred as cone and extending indefinitely far in both directions as in the figure(a).
The point V is called the vertex of the cone and the line l is called the axis of the cone. The rotating line m is called the generator of the cone. The vertex of the cone separates the cone into two parts which is called the nappes.
If a plane intersects the cone, then the session obtained due to the intersection is called a conic section. Thus, conic sections may be defined as the intersection of the right circular cone with a plane.
There obtain a different kinds of the conic sections which depends on the position of the intersecting plane with respect to the cone and by the angle made by it with the vertical axis of the cone. Let ? be the angle made by the intersecting plane with the vertical axis of the cone as shown in figure(c).
The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex.
Cone Intersection
As a plane cuts the nappe (other than the vertex) of the cone, we have the following situations:
(a) If ? = 90?, then the section is a circle [figure (d)].
(b) If ?< ? < 90?, then the section is an ellipse[figure (e)].
(c) If ? = ?; then the section is parabola[figure (f)].
( In each of the these three situations, the plane cuts entirely across one nappe of the cone).
(d) If 0? ?< ?; then the plane cuts through both the nappes and the curves of intersection is hyperbola[figure (g)].
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A parabola may be defined as the set of all points in a plane that are equidistant or are at same distance from a fixed line and a fixed point (not on the line) in the plane.
This fixed line is called the vertex of the parabola and the fixed point F is called the focus of the parabola [figure- (c)].
(Actually ‘Para’ means ‘for’ and ‘bola’ means ‘throwing’, i.e., the shape described when we throw a ball in the air).
A line passing through the focus and is perpendicular to the directrix is called the axis of the parabola. The point of the intersection of the parabola with the axis is called the vertex of the parabola [figure- (d)].
(ii) Ellipse:- An Ellipse imay be defined as set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
These two points are said to be the foci of the ellipse and one point is called the focus of the ellipse[figure- (e)]
The midpoint of the line segment joining both of the focus is called the centre of the ellipse. The line segment passing through the foci of the ellipse is called the major axis and the line segment passing through the centre and is perpendicular to the major axis is called the minor axis. The end points of the major axis are said to be the vertices of the ellipse.[figure- (f)]
A hyperbola is said to be the set of all points in a plane, the difference of whose distance from two fixed points in the plane is a constant. The term ‘difference’ that is used in the definition means the distance to the farther point minus the distance to the closer point. The two fixed points are called the focus of the hyperbola. The mid- point of the line segment joining both of the focus is called the centre of hyperbola. The line passing through the foci is called the transverse axis and the line passing through the centre and is perpendicular to the transverse axis is called the conjugate axis. Where hyperbola intersects the transverse axis, then these points are called the vertices of the hyperbola [figure- (h)].
