Question

lf $a(b-c)x^2+b(c-a)x+c(a-b)$ is a perfect square, then $a,b,c$ are in

• A. A.P.
• B. G.P
• C. H.P.
• D. A.G.P.

Answer 1

Answer: (C)

H.P.

If the equation is a perfect square, it has equal roots
$\Rightarrow b^2=4ac$$\left(b\right(c-a){)}^2=4ac(a-b\left)\right(b-c)$
$b^2(c^2+a^2-2ac)=4ac(ab-b^2-ac+bc)$
$b^2(c^2+a^2+2ac)=4ac(ab-ac+bc)$$b^2(c+a{)}^2=4ac(b(a+c)-ac)$
$\Rightarrow b^2(c+a{)}^2-4a^2{c}^2=0$
$(c+a{)}^2=\frac{4a^2{c}^2}{b^2}$
$\Rightarrow c+a\frac{2ac}{b}$
$\Rightarrow \frac{2}{b}=\frac{1}{c}+\frac{1}{a}$
Hence, $a,b,c$ are in H.P.