Conic Sections

In earlier chapter we have discussed Straight Lines. In this chapter we discuss about some curved lines referred as conic section. A conic section(or simply conic) is a curve obtained by intersection of the surface of a cone with a plane. Here, we discuss about the Important Conic section like Circle, Hyperbola, Parabola, and Ellipse.

The conic sections are produced by intersection of place with a nappe of a cone in following ways (Nappe means either side of a Double Cone: F-2)

• Circle : Circle is produced by in a plane perpendicular to the axis of the cone (F-3)
• Ellipse / Parabola : Plane that is not perpendicular to the axis and intersects a single nappe (F-3)
• Hyperbola: Plane intersecting both nappes (F-4)

Circle

A circle is the set of all points in a plane equidistant from a Fixed Point in the plane (called Centre of the Circle). The distance from the centre to a point on the circle is called the radius of the circle

Equation of a Circle

Parabola

A Parabola is the set of all points (locus) in a plane equidistant from a fixed line and a fixed point (not on the line) in the plane.

• Directrix  : The fixed line is called the directrix of the parabola
• Focus : The fixed point F is called the focus.
• Axis: The line through the focus and perpendicular to the directrix is called the axis of the parabola.
• Vertex: The point of intersection of parabola with the axis is called the Vertex of the parabola.
• Axis of Symmetry : The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the Axis of Symmetry.
• Focal length : The distance between the vertex and the focus, measured along the axis of symmetry, is known as  Focal length.
• Latus Rectum : The chord of the parabola parallel to the directrix and passing through Focus is called Latus Rectum.

The complete parabola has no endpoints. It can extend infinitely to the left, right, and upward.

Ellipse

Ellipse is the set of all points (locus) in a plane, the sum of whose distances from two fixed points in the plane is a constant.

• Centre : The midpoint {\displaystyle C}of the line segment joining the foci is called the center of the ellipse.
• Major Axis : The line through the foci is called the major axis,
• Minor Axis : and the line perpendicular to it through the center is the minor axis.
• Vertex Points :The major axis intersecting the ellipse are known as vertex points
• Focal Distance  : The distance {\displaystyle c}of the foci to the center is called the focal distance 
• Eccentricity : Eccentricity (denoted by e)of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci  (c)  and to one of the vertices (a) of the ellipse. So, e=c/a

Hyperbola

A hyperbola is the set of all points in a plane (locus), the difference of whose distances from two fixed points in the plane is constant. The term ‘Difference’ means the distance to the farther point minus the distance to the closer point.

• Foci: The two fixed points are called the foci of the hyperbola.
• Centre: The mid-point of the line segment joining the foci is called the centre of the hyperbola.
• Transverse Axis : The line through the foci is called the transverse axis
• Conjugate Axis : the line through the centre and perpendicular to the transverse axis is called the conjugate axis.
• Vertices : The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola