A conic section is a curve obtained as the intersection of a cone with a plane. You can obtain different kinds of conic sections depending on the position of the intersecting plane with respect to the cone and by the angle (β) made by it with the vertical axis of the cone.

Conic section consists of those points whose distances to some point (called focus) and some line (called directrix) are in a fixed ratio, called the eccentricity. Three types of conic section are:

  1. Parabola
  2. Ellipse
  3. Hyperbola

Parabola

A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. The fixed line is called the directrix of the parabola and the fixed point F is called the focus. A line through the focus and perpendicular to the directrix is called the axis of the parabola.

Standard Equation

y2 = 4ax; Focus is at F(a,0) and directrix x=-a.

Latus Rectum

Latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola.

Length of the latus rectum = 4a

Ellipse

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci of the ellipse. The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse.

Length of the major axis = 2a

Length of the minor axis = 2b

Standard Equation

x2/a2 + y2/b2 = 1

Latus Rectum

Latus rectum of an ellipse is a line segment perpendicular to the major axis through any of the foci and whose end points lie on the ellipse.

Length of latus rectum = 2b2/a

Eccentricity

The eccentricity of an ellipse is the ratio between the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.

Hyperbola

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant.  The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola.

Standard Equation

x2/a2 - y2/b2 = 1

Latus Rectum

Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola.

Length of latus rectum = 2b2/a

Eccentricity

The eccentricity of a hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola.