Prove that the expression ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 can be resolved into two linear rational factors if Δ = abc + 2fgh − af^2 − bg^2 − ch^2 = 0


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Question Number 20297 by Tinkutara last updated on 25/Aug/17

$Prove that the expression {a x}^{2} + 2 h x y$ $+ {b y}^{2} + 2 g x + 2 f y + c = 0 can be$ $resolved into two linear rational factors$ $if Δ = a b c + 2 f g h - {a f}^{2} - {b g}^{2} - {c h}^{2} = 0$
	Answered by ajfour last updated on 25/Aug/17

	$l e t a {(x - x_{0})}^{2} + b {(y - y_{0})}^{2} +$ $2 h (x - x_{0}) (y - y_{0}) = {a x}^{2} + {b y}^{2} + 2 h x y$ $2 g x + 2 f y + c = 0$ $c o m p a r i n g c o e f f i c i e n t s o f x a n d$ $y, a n d c o n s t a n t t e r m w e g e t :$ ${a x}_{0} + {h y}_{0} = - g . . . . . (i)$ ${b y}_{0} + {h x}_{0} = - f . . . . . . (i i)$ ${a x}_{0}^{2} + {b y}_{0}^{2} + 2 {h x}_{0} y_{0} = c (i i i)$ $u s i n g (i) a n d (i i) i n (i i i) :$ $\Rightarrow x_{0} (- g - {h x}_{0}) + y_{0} (- f - {h x}_{0})$ $+ 2 {h x}_{0} y_{0} = c$ $o r {g x}_{0} + {f y}_{0} = - c . . . . (i v)$ $(i), (i i), a n d (i v) f o r m a s y s t e m$ $o f t h r e e l i n e a r e q u a t i o n s i n t w o$ $u n k n o w n s, h a s u n i q u e s o l u t i o n o n l y i f$ $\| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} \| = 0$ $f i r s t r o w w e g e t f r o m c o e f f i c i e n t s$ $o f x, y, a n d c o n s t a n t t e r m o f$ $e q u a t i o n (i), s e c o n d r o w \to (i i)$ $t h i r d r o w \to (i v)$ $\Rightarrow a (b c - f^{2}) - h (c h - g f) + g (h f - b g) = 0$ $\Rightarrow a b c + 2 f g h - {a f}^{2} - {c h}^{2} - {b g}^{2} = 0 .$
	Commented by Tinkutara last updated on 25/Aug/17

	$Thank you very much Sir!$
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