Two lines together (whether they are parallel, intersect each other or coincide) are called Pair of straight lines.
Combined Equation Of Pair Of Lines:
           If the equations of two straight lines separately are L₁ = 0 and L₂ = 0, then the
combined equation of that pair of lines is given by L₁ L₂ = 0
Reason: (i) All the points of L₁ = 0, satisfy the equation L₁ L₂ = 0
(ii) All the points of L₂ = 0, satisfy the equation, L₁ L₂ = 0
(iii) Those points which are not on L₁ = 0 or L₂ = 0, donot satisfy the equation L₁ L₂ = 0
eg: 1. The combined equations of the lines 2x + 3y + 5 = 0 and 3x - 4y + 7 = 0 is
           (2x + 3y + 5)  (3x - 4y + 7) = 0
2. The combined equation of the lines 4x + y = 0 and 2x + 5y = 0 is (4x + y) (2x + 5y) = 0
Result: The combined equation of two lines
L₁ = l₁x + m₁y = 0
L₂ = l₂x + m₂y = 0,

the equation ax² + 2hxy + by² = 0 represents pair of (real) lines
Important Result: If the scopes of the two (non - vertical) lines represented by ax² + 2hxy + by² = 0 are m₁, m₂ respectively, then we have

Angle between pair of lines: If the equation ax² + 2hxy + by² = 0 represents pair of lines and 'θ' is acute angle between them,

Note: 1. When θ = 90°, a + b = 0
2.  When θ  =  0°,  h² = ab
Bisectors of Angles:
Result: We can show that, the equations to the bisectors of angles between the lines
a₁x + b₁y + c₁ = 0
And a₂x + b₂y + c₂ = 0 are

eg: Find the bisectors of angles between the two given lines
3x + 4y + 1 = 0 and 5x + 12y + 3 = 0
Sol: The equations of bisectors of angle are 

 By simplifying, we can get the equations of bisectors of the angles between given pair of lines.
Important Result: The combined equation of pair of bisectors of angle between the pair of lines ax² + 2hxy + by² = 0 is given by
h(x² − y²) − (a − b)xy = 0
5555555555555555555

Important Results:
1. If the equation S ≡ ax² + 2hxy + by² + 2gx + 2fg + c = 0 represents a pair of lines, then
(i) ∆ ≡ abc + 2fgh − af2 − bg² − ch² = 0
              And
(ii) f² ≥ bc, g2 ≥ ca, h² ≥ ab
2. If S ≡ ax² + 2hxy + by² + 2gx + 2fy + c = 0
represents a pair of parallel lines, then
(i) h² = ab
(ii) af² = bg²

3. The point of intersection of fair of lines,
S ≡ ax² + 2hxy + by² + 2gx + 2fy + c = 0

Curves represented by the general equations of 2nd degree in x, y:
         Depending on some conditions, the general equation of 2nd degree in x, y
         S ≡ ax² + 2hxy + by² + 2gx + 2fy + c = 0
         may represent, a pair of lines or a circle or a conic or may not represent any real point also.
        Now, we will discuss the method of finding out the equation to the pair of lines formed by joining the origin to the points of intersection of given curve ax² + 2hxy + by² + 2gx + 2fy + c = 0 and a line lx + my = 1

Since the required pair of lines pass through origin and satisfy the common point of given curve and lines. We get the equation of required pair of lines by homogenising the curve using the given line.
eg: Find the combined equation of pair of lines, obtained by joining the origin to the point of intersection of x² + y² = 1 and x + y = 1.
Solution: