The value of 'a' for which the lines 2ax - 2y + 3z = 0; x + ay + 2z = 0 and 2x + az = 0 are concurrent, is .

-2

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-1

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Solution

The correct option is B 2
If the lines are concurrent, then
$⎛ ⎜ ⎝ \begin{matrix} 2 a & - 2 & 3 1 & a & 2 2 & 0 & a \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow 2 a (a^{2} - 0) + 2 (a - 4) + 3 (0 - 2 a) = 0 \Rightarrow 2 a^{3} + 2 a - 8 + 0 - 6 a = 0 \Rightarrow 2 a^{3} - 4 a - 8 = 0 \Rightarrow a^{3} - 2 a - 4 = 0 \Rightarrow a^{3} - 2 a^{2} + 2 a^{2} - 4 a + 2 a - 4 = 0 \Rightarrow a^{2} (a - 2) + 2 a (a - 2) + 2 (a - 2) = 0 \Rightarrow (a - 2) (a^{2} + 2 a + 2) = 0 \Rightarrow a = 2$

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