1. ax² + 2hxy + by² = 0 is called a homogeneous equation of 2nd degree 

    passing through the origin.

 

a) two lines r real/distinct if h² > ab

 

b) two lines r imaginary if  h² < ab

 

c) two lines r coincident if h² = ab 

 

 

2. If ax² + 2hxy + by² = 0 = b(y - m₁x)(y - m₂x)

   

    then m₁ + m₂ = -2h/b   & m₁m₂ = a/b

 

 

3. Angle b/w two above lines is theta = tan^-12h² - ab/ a + b

   

    a)  theta = 90^o a+ b = 0           b) theta = 0^o if h² = ab

 

 

4. Equation of pair of straight lines perpendicular to ax² + 2hxy + by² = 0 is

   

     bx² - 2hxy + ay²

 

 

5. Equation of bisectors of ax² + 2hxy + by² = 0  is (x² - y²)/a - b = xy/h

   

    if a = b then bisectors  : y =  x

    

    if h = 0,then bisectors  : x = 0, y = 0

 

 

6. General equation of 2nd degree ax² + 2hxy + by² +2gx + 2fy + c = 0        

 

    represents a pair of straight lines if

 

    abc + 2fgh - a f² - bg² - ch² = 0

 

 

7. The angle b/w above two lines is theta =   tan^-12h² - ab/ a + b

 

 

    a) lines parallel : h² = ab

 

    b) lines perpendicular : a + b = 0

    

    c) lines coincident if h² = ab , g² = ac , f² = ac

 

 

8. Pt. of intersection of lines S = ax² + 2hxy + by² +2gx + 2fy + c = 0 can b 

 

    obtained by solving

 

    delta s/delta x = 0  & delta s/delta y = 0

 

 

9. If (x₁,y₁) is the point of intersection of such lines then the eqn of bisectors of

 

   S = 0 is

 

   [(x - x₁)^₂ - (y - y₁)²] a - b  = (x - x₁)(y - y₁)/h