pencil of conics

Two conics (

U= 0andV= 0 (1)

can intersect in four points, some of which may coincide or be “imaginary”.

The equation

pU+qV= 0, (2)

where p and q are freely chooseable parametres, not both 0, represents the pencil of all the conics which pass through the four intersection points of the conics (1); see quadratic curvesMathworldPlanetmath.

The same pencil is gotten by replacing one of the conics (1) by two lines  L1=0  and  L2=0,  such that the first line passes through two of the intersection points and the second line through the other two of those points; then the equation of the pencil reads

pL1L2+qV= 0. (3)

One can also replace similarly the other (V) of the conics (1) by two lines  L3=0  and  L4=0; then the pencil of conics is

pL1L2+qL3L4= 0. (4)

For any pair  (p,q)  of values, one conic sectionMathworldPlanetmath (4) passes through the four points determined by the equation pairs


The pencils given by the equations (2), (3) and (4) can be obtained also by fixing either of the parametres p and q for example to -1, when e.g. the pencil (4) may be expressed by

pL1L2=L3L4. (5)

Application.  Using (5), we can easily find the equation of a conics which passes through five given points; we may first form the equations of the sides  L1=0,  L2=0,  L3=0  and  L4=0  of the quadrilateralMathworldPlanetmath determined by four of the given points.  The equation of the searched conic is then (5), where the value of p is gotten by substituting the coordinates of the fifth point to (5) and by solving p.

Example.  Find the equation of the conic section passing through the points

(-1, 0),(1, 0),(0, 1),(0, 2),(2, 2).

We can take the lines


passing through pairs of the four first points.  The equation of the pencil of the conics passing through these points is thus of the form

p(2x+y-2)(x-y+1)=(2x-y+2)(x+y-1). (6)

The conics passes through  (2, 2), if we substitute  x:=2,  y:=2;  it follows that  p=3.  Using this value in (6) results the equation of the searched conics:

2x2-y2-2xy+3y-2= 0 (7)

The coefficients 2, -1, -2 of the second degree terms let infer, that this curve is a hyperbolaMathworldPlanetmath with axes not parallelMathworldPlanetmathPlanetmath to the coordinate axes (see quadratic curves (

Title pencil of conics
Canonical name PencilOfConics
Date of creation 2013-03-22 18:51:07
Last modified on 2013-03-22 18:51:07
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 21
Author pahio (2872)
Entry type Definition
Classification msc 51N20
Classification msc 51A99
Related topic QuadraticCurves
Related topic LineInThePlane