Given lines are and l+m+n=0 . . . (i) mn−2ln+lm=0 . . . (ii) From equation (i) l=−(m+n) Putting in equation (ii), we get ⇒mn+2(m+n)n−(m+n)m=0 ⇒mn+2mn+2n2−m2−nm=0 ⇒2n2−m2+2mn=0 2(
n
m
)2+
2n
m
−1=0 This is a quadratic equation in (
n
m
). ∴
n1n2
m1m2
=
−1
2
. . . (iii) [where
n1
m1
,
n2
m2
are the roots of the equation] From equaiton (i) m=−(n+l) Putting in equation (ii), we get −(n+l)n−2ln−l(n+l)=0 ⇒n2+ln+2ln+ln+l2=0 ⇒l2+3ln+n2=0 ⇒(
l
n
)2+
3l
n
+1=0 ⇒
l1l2
n1n2
=1 . . . (iv) [where
l1
n1
,
l2
n2
are the roots of the equation] From equations (iii) and (iv), we get l1l2=−
