if 3x^2 +2αxy+2y^2 +2ax−4y+1 can be resolved into two linear factors, prove that ′α′ is a root of the equation x^2 +4ax+2a^2 +6=0


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Question Number 37166 by nishant last updated on 09/Jun/18

$i f 3 x^{2} + 2 α x y + 2 y^{2} + 2 a x - 4 y + 1$ $c a n b e r e s o l v e d i n t o t w o l i n e a r$ $f a c t o r s, p r o v e t h a t^{'} α^{'} i s a r o o t$ $o f t h e e q u a t i o n x^{2} + 4 a x + 2 a^{2} + 6 = 0$
Commented by prakash jain last updated on 10/Jun/18

${a x}^{2} + 2 h x y + {b y}^{2} + 2 g x + 2 f y + c = 0$ $a = 3$ $h = α$ $b = 2$ $g = a$ $f = - 2$ $c = 1$ $Δ = \| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \end{matrix} \|$ $\| \begin{matrix} 3 & α & a \\ α & 2 & - 2 \\ a & - 2 & 1 \end{matrix} \|$ $= - (α^{2} + 4 a α + 2 a^{2} + 6)$ $(α^{2} + 4 a α + 2 a^{2} + 6) = 0$ $i m p l i e s α i s r o o t o f a b o v e e q u a t i o n .$ $This is necessary but not sufficient$ $condition .$ $for the quadratic to represent a pair$ $a straight lines you need$ $a b - h^{2} = 6 - α^{2} < 0$ $Other Cases$ $Δ = 0, a b - h^{2} < 0 \Rightarrow 2 l i n e s$ $Δ = 0, a b - h^{2} > 0 \Rightarrow one point$ $Δ = 0, a b - h^{2} = 0 \Rightarrow one line$
Commented by prakash jain last updated on 10/Jun/18

$You can use the following sentence$ $to memorize the determinant .$ $a ll h ostel g uys h ave b est f riends$ $g o f or c inema$
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