if-3x-2-2-xy-2y-2-2ax-4y-1-can-be-resolved-into-two-linear-factors-prove-that-is-a-root-of-the-equation-x-2-4ax-2a-2-6-0-

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