for x being real when the expression (x-a)(x-b)/(x-c) assumes real values

ques of iit 1994

please give deyailed solution

(x-a)(x-b)/(x-c) =k(real), x²-(a+b)x+ab=kx-kc, x²-(a+b+k)x+ab+kc=0,if x is real, discriminant≥0,(a+b+k)^2≥4(ab+kc)

k²+(a+b)²+2k(a+b)≥4(ab+kc), k²+2k(a+b-2c)+(a-b)^2≥0,k≤-(a+b-2c)-√((a+b-2c)²-(a-b)²)

or k≥-(a+b-2c)+√((a+b-2c)²-(a-b)²)