The given pair of lines 18x2−9xy+y2=0 is homogeneous. ⇒ The two lines passes through the origin, i.e O(0,0). The other two points of the triangle can be determined by substituting y=c in equation of pair of lines 18x2−9xc+c2=0 ⇒ 18x2−6xc−3xc+c2=0 ⇒ 6x(3x−c)−C(3x−c)=0 (6x−c)(3x−c)=0⇒x=
c
6
,x=
c
3
The two points are (
c
6
,c) and (
C
3
,c) Area =27
1
2
|
0
0
1
c
3
c
1
c
6
c
1
|=27⇒
1
2
[
c2
3
−
c2
6
]=27 ⇒
1
2
×
c2
6
=27⇒c2=27×2×6 c=18 The three points of the triangle are (0,0),(3,18),(6,18). Centroid of triangle =(