Question

If the roots of the equation $(b-c)x^2+(c-a)x+(a-b)=0$ are equal, then a,b,c will be in-

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. None of these

Answer 1

The correct option is A $A.P.$

(b-c)x²+(c-a)x+(a-b)=0

Comparing with quadratic equation

Ax²+Bx+C=0

A=(b-c),B=c-a,C=a-b

Discriminant is eqwual to zero when roots are equal

D=B²-4AC=0

D=(c-a)²−4(b-c)(a-b)=0

D=(c²+a²−2ac)-4(ba-ac-b²+bc)=0

D=c²+a²−2ac-4ab+4ac+4b²-4bc=0

c²+a²+2ac-4b(a+c)+4b²=0

(a+c)²-4b(a+c)+4b²=0

[(a+c)-2b]²=0

a+c=2b,

Thus, a , b, c are in A.P.