Question

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

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

Answer 1

The correct option is C

H.P

Since the roots of the quadratic equation in x are equal, we have $(B^2-4AC)$ = 0

$\Rightarrow b^2(c-a{)}^2-4ac(b-c\left)\right(a-b)$ = 0

$\Rightarrow b^2(c^2-2ca+a^2)-4ac(ba-b^2-ca+bc)$ = 0

$\Rightarrow b^2(c^2+2ca+a^2)-4ac\left\{b\right(a+c)-ac\}$ = 0

$\Rightarrow b^2(a+c{)}^2-4ac\{b(a+c)-ac\}$ = 0

Which can be seen to be true, if

b = $\frac{2ac}{a+c}$ or b(a+c) = 2ac i.e., if a,b,c are in H.P.