If planes r→.(i^+j^+k^)= 1, r→.(i^+2aj^+k^)= 2 and r→.(ai^+a2j^+k^)= 3 intersect in a line, then the number of real values of “a” possible is(are) ____

Given equation of planes areP1 : x + y + 1 = 1 …….... (1)P2 : x + 2ay + z = 2 …….... (2)P3 : ax + a2 y + z = 3 …….... (3)If 3 planes intersect in a line, then |11112a1aa21|=0Applying C1→C1−C2 and C2→C2−C3 |0011−2a2a−11a−a2a2−11|=0 On expanding along C1, we get ⇒(1−2a)(a2−1)−(a−a2)(2a−1)=0But, for a = 1, planes P1 and P3 are parallel and for a =12 , planes P1 and P2 are parallel.Hence, we conclude that 3 planes will never intersect in a line for any real value of a.

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If planes $\vec{r} . (\hat{i} + \hat{j} + \hat{k})$ = 1, $\vec{r} . (\hat{i} + 2 a \hat{j} + \hat{k})$ = 2 and $\vec{r} . (a \hat{i} + a^{2} \hat{j} + \hat{k})$ = 3 intersect in a line, then the number of real values of “a” possible is(are) ____

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Detailed Solution

Given equation of planes are
$P_{1}$ : x + y + 1 = 1 …….... (1)
$P_{2}$ : x + 2ay + z = 2 …….... (2)
$P_{3}$ : ax + $a^{2}$ y + z = 3 …….... (3)
If 3 planes intersect in a line, then
$| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 a & 1 \\ a & a^{2} & 1 \end{matrix} | = 0$
Applying $C_{1} \to C_{1} - C_{2}$ and $C_{2} \to C_{2} - C_{3}$
   $| \begin{matrix} 0 & 0 & 1 \\ 1 - 2 a & 2 a - 1 & 1 \\ a - a^{2} & a^{2} - 1 & 1 \end{matrix} | = 0$
On expanding along $C_{1}$ , we get
   $\Rightarrow (1 - 2 a) (a^{2} - 1) - (a - a^{2}) (2 a - 1) = 0$
But, for a = 1, planes    $P_{1}$ and    $P_{3}$ are parallel and for a = $\frac{1}{2}$ , planes    $P_{1}$ and $P_{2}$ are parallel.
Hence, we conclude that 3 planes will never intersect in a line for any real value of a.