Pair of Lines

 

General equation of second degree

The equation of the form ax² + 2hxy + by² + 2gx + 2fy + c = 0 is the general equation second degree in x and y.

The pair of straight lines A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0 can always be put in the form of second degree equation but the converse is not always true.

Homogeneous equation of second degree

The equation of the form ax² + 2hxy + by² = 0 is the called homogeneous equation of second degree in x and y. The homogeneous equation of second degree always represents a pair of straight lines passing through the origin.

Introduction

Consider two equation of straight lines

             a₁x + b₁y + c₁ = 0                                              ….(i)

and        a₂x + b₂y + c₂ = 0                                              …(ii)

The combined equation of the above equation is

             (a₁x + b₁y + c₁) (a₂x + b₂y + c₂) = 0       ….(iii)

The coordinates of any point lying on lines (i) and (ii) will satisfy the equation (iii). Conversely, the coordinates of any point lying on equation (iii) also satisfy (i) or (ii) or both.

Now, on simplification of equation (iii) yields, a₁a₂x² + (a₁b₂ + a₂b₁) xy + b₁b₂ y² + (a₁a₂ + a₂c₁)x + (b₁c₂ + b₂c₁)y + c₁c₂ = 0, which can also be written as

ax² + 2hxy + by² + 2gx + 2fy + c = 0

where, a = a₁a₂, 2h = a₁b₂ + a₂b₁, b = b₁b₂, 2g = a₁c₂+ a₂c_1,

2f = b₁c₂ + b₂c₁ and c = c₁c₂.

Hence every equation of pair of line is a second degree equation.

Note: The equation ax² + 2hxy + by² + 2gx +2fy + c = 0 is called general equation second degree in x and y.

Theorem

The homogeneous equation of second degree in x and y represents a pair straight line passing through the origin.

The homogeneous equation of second degree in x and y is

            ax² + 2hxy + by² = 0.

The equation can be written as

        Angle between Lines Represented by ax²+2hxy+by² = 0

Let,       y = m₁x                                              .....(i)

and,       y = m₂x                                            .....(ii)

be equations of two lines represented by the equation ax² + 2hxy+ by² = 0.

      Bisectors of the angles between the lines represented by

       ax² + 2hxy + by² = 0

       Condition that the General Equation of Second Degree to Represent a Pair of Lines

The general equation of second degree in x and y is;

             ax² + 2hxy + by² + 2gx + 2fy + c = 0 (a ¹ 0)

The equation may be written as a quadratic in x, as

             ax² + 2x(hy + g) + by² + 2fy + c = 0

Solving for x;

These equations will be linear only if (hy + g)² – a(by² + 2fy + c) be perfect square,

i.e.,       (h² – ab) y² + 2(gh – af)y + (g² – ac) be a perfect square

i.e.,       {2(gh – af)²} – 4(h² – ab) (g² – ac) = 0

i.e.,       (gh – af)² – (h² – ab) (g² – ac) = 0

i.e.,       g²h² – 2afgh + a²f² –g²h²+ h²ac + g²ab - a²bc = 0

i.e.,       a(abc + 2fgh – af² – bg² – ch²) = 0

Since a ¹ 0, the required conditions is abc + 2fgh – af² – bg² –  ch² = 0.

If a = 0 but b ¹ 0, then the given equation can be expressed as a quadratic in y and we can find the condition in the same way.

     Equation of the Pair of Lines Joining Origin to the Points of Intersection of Line and Curve

Let        lx + my = n                                                      ….(i)

be a straight line and

             ax² + 2hxy + by² + 2gx + 2fy + c = 0   …(ii)

be the curve.