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Question

If the slope of one of the lines represented by ax2+2hxy+by2 be the nth power of the other, prove that
(abn)1n+1+(anb)1n+1+2h=0

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Solution

This represents pair of straight lines through origin
Let y=m1x y=m2x be two lines
mx mnx
(ymx)(ymnx)=0
y2(mn+m)xy+mn+1x2=0
a=1 b=mn+1 h=(mn+m)2
=(abn)1n+1+(anb)1n+1+2h
=mn+m+22(mnm)
=mnmn+mm
=0+0
=0

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