Question

If $a(b-c)x^2+b(c-a)xy+c(a-b)y^2$ is a perfect square, then the quantities $a,b,c$ are in

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

Answer 1

Answer: (C)

H.P

Since, $a(b-c)x^2+b(c-a)xy+c(a-b)y^2$ is a perfect square.
Therefore the roots of $a(b-c)x^2+b(c-a)xy+c(a-b)y^2=0$ are equal.
$\therefore D={\left[b\right(c-a\left)\right]}^2-4ac(b-c)(a-b)=0$
$\Rightarrow b^2{(c-a)}^2=-4ac[b^2-b(a+c)+ac]$
$\Rightarrow b^2[{(c-a)}^2+4ac]-4abc(a+c)+4a^2{c}^2=0$
$\Rightarrow b^2{(c+a)}^2-4abc(a+c)+4a^2{c}^2=0$
$\Rightarrow {[b(a+c)-2ac]}^2=0$
$\Rightarrow b(a+c)=2ac$
$\Rightarrow b=\frac{2ac}{a+c}$
Therefore, $a,b,c$ are in H.P
Ans: C