For a,b,c non-zero, real distinct, the equation, a2+b2x2−2b(a+c)x+b2+c2=0 has non-zero real  roots. One of these roots is also the root of the equation:

a2+b2x2−2b(a+c)x+b2+c2=0D=4b2(a+c)2−4a2+b2b2+c2 =−4b4−2b2ac+a2c2=−4b2−ac2 For real roots, D≥0⇒ −4b2−ac2≥0⇒ b2−ac=0⇒ Roots are real and equal. ∴ Roots are  2b(a+c)±02a2+b2 =b(a+c)a2+ac =baThis root satisfies option (c).