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

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 \dots . . (i)

{b y}_{0} + {h x}_{0} = - f \dots \dots (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 \dots . (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!