define family of lines

define family of lines


3 Answers

AKASH GOYAL AskiitiansExpert-IITD
420 Points
12 years ago

Dear Sushant

any line through the point of intersection of the lines a1x +b1y +c1=0 and a2x +b2y +c2=0 can be represented by the equation

(a1x +b1y +c1) + λ(a2x +b2y +c2)=0

where λis a parameter.

this the family of straight lines

All the best.




Please feel free to post as many doubts on our discussion forum as you can. We are all IITians and here to help you in your IIT JEE preparation.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar  respectively : Click here to download the toolbar..

pawan deep
41 Points
12 years ago
In solid geometry, a family of Lines may actually be curved. For example a line may be drawn on the surface of a sphere. In plane geometry, we describe it as totally straight in all dimensions.

        If a Family of Lines is not straight, we usually refer to it as a curvature, arc, or curved line. In flat surface geometry the word 'line' is usually taken to mean a straight line.
Concept of Family of Lines:

       The joint equation of the straight lines a1x +b1y +c1 = 0 and a2x +b2y+ c2 = 0 is

                    (a1x +b1y +c1) (a2x +b2y+ c2) = 0

                    ax² +2hxy+by²+2gx+2fy+c = 0 is joint equation of a pair of straight lines(family of lines).


                                         The equation ax² +2hxy+by²+2gx+2fy+c = 0 is known as general equation of second degree.

                                         The equation ax² +2hxy+by² = 0 is known as homogeneous equation of second degree.

         In a homogeneous equation of second degree, the sum of indices (exponents) of x and y in each term is equal to 2.

         In family of lines, the homogeneous equation of second degree ax² +2hxy+by² = 0 represents a joint equation of two straight lines passing through the origin if h²≥ab.

          If y = m1x and y = m2x are the lines represented by a homogeneous equation of second degree ax² +2hxy+by² = 0, then

                        (i) m1 =m2 = -2h/b

                        (ii) m1m2 = a/b

           The angle θ between the pair of lines represented the homogeneous equation of second degree ax² +2hxy+by² = 0 is given by

                            tan θ = [2√(h² –ab)]/(a+b)

                      If θ = 0, which means h² = ab lines are coincident.

                                     Lines are perpendicular (Family of Lines) means θ = π/2, tan θ = ∞, and cot θ = 0.This means a+b = 0 or a = -b
              Coefficient of x² = coefficient of y²

                   ax² +2hxy+by²+2gx+2fy+c = 0 will represent a pair of straight lines if the determinant

                             [[[a,h,g],[h,b,f],[g,f,c]]] = 0

              Expanding the determinant 

                           abc +2fgh -af² -bg² -ch² = 0

               Angle θ between the lines represented by the general second degree equation ax² +2hxy+by²+2gx+2fy+c = 0 is given by

tan θ = [2 √(h² – ab)]/(a+b)
Algorithm to Find Separate Equations of Family of Lines:

Algorithm to find separate equations of lines in ax² +2hxy+by²+2gx+2fy+c = 0

         Step 1: 

                Find factors for the homogeneous part ax² +2hxy+by². Let the factors be
                                   (a1x +b1y ) and (a2x +b2y )

         Step 2:

                 Add constants c1 and c2 to them. (a1x +b1y +c1) and (a2x +b2y +c2).

         Step 3:

                 Multiply (a1x +b1y +c1) and (a2x +b2y +c2) and compare with ax² +2hxy+by²+2gx+2fy+c to obtain equations in c1 and c2.

         Step 4:

                 Solve the equations and get values of c1 and c2.

Pls. Do Reply.......
Radhika Batra
247 Points
10 years ago

What is the interpretation of the parameter LAMBDA?

Think You Can Provide A Better Answer ?


Get your questions answered by the expert for free