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Question
Prove that one of the lines represented by the equation
ax3+bx2y+cxy2+dy3=0
will bisect the angle between the other two if
(3a+c)2(bc+2cd−3ad)=(b+3d)2(bc+2ab−3ad).
ax3+bx2y+cxy2+dy3=0
will bisect the angle between the other two if
(3a+c)2(bc+2cd−3ad)=(b+3d)2(bc+2ab−3ad).
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Solution
ax3+bx2y+cxy2+dy3=0
Lines
represented by the equation will be of form y=mx
⇒dm3+cm2+bm+a=0........(i)m1+m2+m3=−cd.......(ii)m1m2m3=−ad.......(iii)
One
of the line is the angle bisector
m2−m11+m2m1=m3−m21+m2m3m2+m22m3−m1−m1m2m3=m3+m1m2m3−m2−m22m12m2+m22(m3+m1)=2m1m2m3+m3+m1
Using
(ii) and (iii)
2m2+m22(−cd−m2)=2−ad−m2−cd3m2−m32−cdm22=−2a+cd3dm2−dm32−cm22=−2a−cdm32+cm2=2a+c+3dm2......(iv)
m2 is also a root of (i)
⇒dm32+cm22+bm2+a=0dm32+cm22=−bm2−a........(v)
From
(iv) and (v)
−bm2−a=2a+c+3dm23dm2+bm2=−3a−cm2=−3a−c3d+b=−3a+c3d+b
Substituting m2 in (i)
−d(3a+c3d+b)3+c(3a+c3d+b)2−b3a+c3d+b+a=0(3a+c3d+b)2(c−d3a+c3d+b)=b3a+c3d+b−a(3a+c3d+b)2(3cd+bc−3ad−cd)=3ab+bc−3bd−ab(3a+c)2(2cd+bc−3ad)=(3d+b)2(2ab+bc−3bd)
Hence proved.
ax3+bx2y+cxy2+dy3=0
Lines represented by the equation will be of form y=mx
⇒dm3+cm2+bm+a=0........(i)m1+m2+m3=−cd.......(ii)m1m2m3=−ad.......(iii)
One of the line is the angle bisector
m2−m11+m2m1=m3−m21+m2m3m2+m22m3−m1−m1m2m3=m3+m1m2m3−m2−m22m12m2+m22(m3+m1)=2m1m2m3+m3+m1
Using (ii) and (iii)
2m2+m22(−cd−m2)=2−ad−m2−cd3m2−m32−cdm22=−2a+cd3dm2−dm32−cm22=−2a−cdm32+cm2=2a+c+3dm2......(iv)
m2 is also a root of (i)
⇒dm32+cm22+bm2+a=0dm32+cm22=−bm2−a........(v)
From
(iv) and (v)
−bm2−a=2a+c+3dm23dm2+bm2=−3a−cm2=−3a−c3d+b=−3a+c3d+b
Substituting m2 in (i)
−d(3a+c3d+b)3+c(3a+c3d+b)2−b3a+c3d+b+a=0(3a+c3d+b)2(c−d3a+c3d+b)=b3a+c3d+b−a(3a+c3d+b)2(3cd+bc−3ad−cd)=3ab+bc−3bd−ab(3a+c)2(2cd+bc−3ad)=(3d+b)2(2ab+bc−3bd)
Hence proved.
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