Question

If the lines $\{\overline{r}-(2\hat{i}-3\hat{j}+4\hat{k}\left)\right\}.\left\{\right(1-p)\hat{i}+3\hat{j}-4\hat{k}\}=0$ and $\{\overline{r}-(3\hat{i}-4\hat{j}+5\hat{k}\left)\right\}.\{2\hat{i}-4\hat{j}+5\hat{k}\}=0$ are coplanar, then the value of $p$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

The first line passes through $2\hat{i}-3\hat{j}+4\hat{k}$ and perpendicular to $(1-p)\hat{i}+3\hat{j}-4\hat{k}$
The second line passes thought $3\hat{i}-4\hat{j}+5\hat{k}$ and perpendicular to $2\hat{i}-4\hat{j}+5\widehat{k.}$

Hence, the condition for the planes to be coplanar is
$\left|\begin{array}{ccc}3-2& -4+3& 5-4\\ (1-p)& 3& -4\\ 2& -4& 5\end{array}\right|=0$

$\Rightarrow p=2$