PAIR OF STRAIGHT LINES
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
-12
h
2 - ab/ a + b
a) theta = 90o a+ b = 0 b) theta = 0o 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
-12
h
2 - 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
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