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:
b2−c2x2+2a(b−c)x−a2=0
b2+c2x2−2a(b+c)x+a2=0
a2x2+a(c−b)x−bc=0
a2x2-a(c−b)x+bc=0
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 =ba
This root satisfies option (c).