If-the-system-of-equations-ax-by-a-b-z-0-bx-cy-b-c-z-0-a-b-x-b-c-y-0-has-a-non-trivial-solution-then-

Question Number 56422 by gunawan last updated on 16/Mar/19

If the system of equations

a x + b y + (a λ + b) z = 0

b x + c y + (b λ + c) z = 0

(a λ + b) x + (b λ + c) y = 0

has a non - trivial solution, then

Answered by MJS last updated on 17/Mar/19

solving we get

x = \frac{(a λ + b) (b λ + c)}{b^{2} - a c} z

y = - \frac{{(a λ + b)}^{2}}{b^{2} - a c} z

(a λ^{2} + 2 b λ + c) z = 0

z = 0 \Rightarrow x = y = 0

a λ^{2} + 2 b λ + c = 0 \Rightarrow λ = - \frac{b}{a} \pm \frac{\sqrt{b^{2} - a c}}{a}

\Rightarrow (\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} (\frac{b}{a} \pm \frac{\sqrt{b^{2} - a c}}{a}) p \\ - p \\ p \end{matrix})

with p \in R \land a \neq 0 \land b^{2} ⩾ a c

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