Question

The condition such that $a{x}^2+2cxy+b{y}^2+2bx+2ay+c$ is resolvable into two linear factors is

• A. $a^3+b^3+c^3=3abc$
• B. $a^3+b^3+c^3=abc$
• C. $a^3+b^3+c^3=ab+bc+ca$
• D. $a^3+b^3+c^3+abc=0$

Answer 1

The correct option is A $a^3+b^3+c^3=3abc$

The condition for $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0.$
is $\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0.$ to represent a pair of lines that is to be resolved into factor.
$\therefore \Delta =abc+2\left(c\right)\left(b\right)\left(a\right)-a(a{)}^2-b(b{)}^2-c(c{)}^2=0$
$\Rightarrow a^3+b^3+c^3=3abc$