If the planes r→⋅(i^+j^+k^)=q1, r→⋅(i^+2aj^+k^)=q2 and r→⋅ai^+a2j^+k^=q3 intersect in a line, then the sum of all values of 'a' is ________

r→⋅n→1=q→1 and r→⋅n→2=q→2,r→⋅n→3=q→3 intersect in a line if n→1 n→2 n→3=0. So 11112a1aa21=0 ⇒1(2a-a2)-1(1-a)+1(a2-2a2)=0 ⇒2a-a2-1+a-a2=0 ⇒2a2-3a+1=0 ∴sum of values of a=32=1.5

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If the planes $\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = q_{1}, \vec{r} \cdot (\hat{i} + 2 a \hat{j} + \hat{k}) = q_{2} and \vec{r} \cdot (a \hat{i} + a^{2} \hat{j} + \hat{k}) = q_{3}$ intersect in a line, then the sum of all values of 'a' is ________

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answer is 1.5.

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Detailed Solution

$\vec{r} \cdot {\vec{n}}_{1} = {\vec{q}}_{1} and \vec{r} \cdot {\vec{n}}_{2} = {\vec{q}}_{2}, \vec{r} \cdot {\vec{n}}_{3} = {\vec{q}}_{3}$ intersect in a line if $[\begin{array}{lll} {\vec{n}}_{1} {\vec{n}}_{2} {\vec{n}}_{3} \end{array}] = 0$ . So $|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 a & 1 \\ a & a^{2} & 1 \end{array}| = 0 \Rightarrow 1 (2 a - a^{2}) - 1 (1 - a) + 1 (a^{2} - 2 a^{2}) = 0 \Rightarrow 2 a - a^{2} - 1 + a - a^{2} = 0 \Rightarrow 2 a^{2} - 3 a + 1 = 0 ∴ s u m o f v a l u e s o f a = \frac{3}{2} = 1.5$