Question

If the planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=q_1,\overrightarrow{r}\cdot (\hat{i}+2a\hat{j}+\hat{k})=q_2$ and $\overrightarrow{r}\cdot (a\hat{i}+a^2\hat{j}+\hat{k})=q_3$ intersect in a line, then the value of $a$ is-

• A. $1$
• B. $\frac{1}{2}$
• C. $2$
• D. $0$

Answer 1

Answer: (A), (B)

The correct options are
A $1$
B $\frac{1}{2}$

Given plane intersects on a line if,
$\left|\begin{array}{ccc}1& 1& 1\\ 1& 2a& 1\\ a& a^2& 1\end{array}\right|=0$
Operating $C_1\to C_1-C_2$ and $C_2\to C_2-C_3$
$\Rightarrow \left|\begin{array}{ccc}0& 0& 1\\ 1-2a& 2a-1& 1\\ a-a^2& a^2-1& 1\end{array}\right|=0$
$\Rightarrow (1-2a)(a^2-1)-(2a-1)(a-a^2)=0$
$\Rightarrow (a-1)(2a-1)=0$
$\Rightarrow a=1$ or $\frac{1}{2}$