The equation A²/(x-a)+B²/(x-b)+C²/(x-c)+....+H²/(x-h)=k,has

(a)no real root

(b)at the most one real root

(c) no complex root

(d) at the most 2 complex roots

The option is C , it has all real roots i.e., no complex roots

T o prove this

lets take (p + iq) as a root of above eqn, the for x=(p + iq)

we have

A²/((p + iq)-a) + B²/((p + iq)-b) + ...............+ H²/((p + iq)-h)   - k  = 0

we can write  this eqn in simplified form as

A².[((p -a)- iq)) / ((p-a)²+q²)] + B².[((p -b)- iq)) / ((p-b)²+q²)] + ...............+ H².[((p -h)- iq)) / ((p-h)²+q²)]   - k  = 0

equating imaginery partson both sides to 0 , we have

q* {( A² / ((p-a)²+q²)+B²/ ((p-b)²+q²)+ .................+H² / ((p-h)²+q²)}  = 0

so q = 0           [since A^{2 ,} B² ,............etc are all positive ]

hence all roots are REAL