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.

AKASH GOYAL

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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).

                                Where

                                         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.


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