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z^n=(z+1)^n. roots of the equation lie on the line
a) 2x+1=0
b)2x-1=0
c)x+1=0
d)x-1=0
The answer is A
let w=(z+1)/z , then wn=1 and w not equal to 1
w = [ cos(2rπ)+i sin(2rπ) ]1/n
= cos(2rπ/n)+i sin(2rπ/n) ( de moivres theorem )
w = cos(2rπ/n)+i sin(2rπ/n) where r=1,2,3,.......n-1
(z+1)/z = cos(2rπ/n)+i sin(2rπ/n)
1/z = -1+ cos(2rπ/n)+i sin(2rπ/n)
= -2 sin2(rπ/n) + 2i sin(rπ/n).cos(rπ/n)
= 2i sin(rπ/n)[ cos(rπ/n)+i sin(rπ/n)]
=> x +iy = z = 1 / {2i sin(rπ/n).[ cos(rπ/n)+i sin(rπ/n)]}
= [ cos(rπ/n)+i sin(rπ/n)]-1 / [2i sin(rπ/n)]
= [ cos(rπ/n) -i sin(rπ/n)] / [2i sin(rπ/n)] ( De moivres theorem )
= -1/2 - i.cot(rπ/n)/2 where r = 1,2,3,.......n-1
this shows that all roots lie on 2x+1=0
Any solution to the equation zn = (z+1)n satisfies |z| = |z+1| i.e. the point is equidistant from (0,0) and (-1,0).
This means it lies on the perpendicular bisector of the line joining (0,0) and (-1,0) i.e. x = -1/2 or 2x+1 = 0.
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