Question

lf $a(b-c)x^2+b(c-a)xy+c(a-b)y^2$ 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.

Considering $y=1$ as constant. The expression becomes a quadratic in $x$.
If it forms a perfect square, then product of roots should be 1, since
$x=1$ is a solution.
Hence,
$\frac{c(a-b)}{a(b-c)}=1$
$ac-bc=ab-ac$
$2ac=ab+bc$
$\frac{2}{b}=\frac{1}{c}+\frac{1}{a}$
Hence,
$\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P
Hence,
$a,b,c$ are in H.P.