Conic Sections

Introduction:
Conic sections, or simply conics, are the curves obtained by cutting a (double) right circular cone by a plane. Let RS be a line through the centre C of a given circle and perpendicular to its plane. Let A be a fixed point on RS. All lines through A and points on the circle generate a right circular cone. The lines are called rulings or generators of the cone. The surface generated consists of two parts, called nappes, meeting at the fixed point A, called the vertex or apex of the cone. The line RS is called axis of the cone.
If the cone is cut by a plane perpendicular to the axis of the cone, the the section is a circle.

The size of the circle depends on how near the plane is to vertex of the cone. If the plane passes through the vertex A, the intersection is just a single point or a  point circle. If the cutting plane is slightly tilted and cuts only one nappe of the cone, the resulting section is an ellipse. If the intersecting plane is parallel to a generator of the cone, but intersects its one nappe only, the curve of intersection is a parabola. If the cutting plane is parallel to the axis of the cone and intersects both of its nappes, then the curve of intersection is a hyperbola.

Equation Of a Circle:

Circle: The set of all points in the plane that are equally distant from a fixed point is called a circle. The fixed point is called the centre of the circle.

Radius: The distance from the centre of the circle to any point on the circle is called the radius of the circle.

Equation Of The Circle In Standard Form:

(x - h)² + (y - k)² = r²     ..................(i)

Point Circle:

If the centre of the circle is the origin, then (i) reduces to 

x² + y² = r²                    ...................(ii)

If r = 0, the circle is called a point circle which consists of the centre only.