Statement-1: Consider a general expression of degree 2 in two variables as $f(x, y)=5 x^2+2 y^2-2 x

Question

Statement-1: Consider a general expression of degree 2 in two variables as $f(x, y)=5 x^2+2 y^2-2 x y-6 x-6 y+9$ can be resolved into two linear factors over real coefficients \Statement-2 : If we compare $f(x, y)$ with $a x^2+b y^2+2 h x y+2 g x+2 f y+c=0$ we have $a b c+2 f g h-a f^2-b g^2-c h^2=0$

Professor PC Thomas Classes · Answer

$\Delta=0 \Rightarrow f(x,y)=5x^{2}+2y^{2}-2xy-6x-6y+9$ can be represented in product of two linear factors. Check if $k(x)=5x^{2}+2y^{2}-2xy$ can be factorized. $2m^{2}-2m+5=0 \Rightarrow D<0$. Factors will not have real coefficients.