Question

$P$ is a point on the line through the point $A$whose position vector is $\overrightarrow{a}$ and the line is parallel to the vector $\overrightarrow{b}$ . If $|\overrightarrow{PA}|=6$, the position vector of $P$ is

• A. $\overrightarrow{b}\pm \frac{3}{|\overrightarrow{a}|}\overrightarrow{a}$
• B. $\overrightarrow{b}\pm \frac{6}{|\overrightarrow{a}|}\overrightarrow{a}$
• C. $\overrightarrow{a}\pm \frac{6}{|\overrightarrow{b}|}\overrightarrow{b}$
• D. $\overrightarrow{a}\pm \frac{3}{|\overrightarrow{b}|}\overrightarrow{b}$

Answer 1

The correct option is C $\overrightarrow{a}\pm \frac{6}{|\overrightarrow{b}|}\overrightarrow{b}$

Let the position vector of point $P$ is $\overrightarrow{r}$ on the line passing through $\overrightarrow{a}$
Since $PA=6$
So, $|\overrightarrow{r}-\overrightarrow{a}|=6\cdots \left(i\right)$
and $PA\parallel \overrightarrow{b}$
So, $\overrightarrow{r}-\overrightarrow{a}=\lambda \overrightarrow{b}\cdots \left(ii\right)$
Hence, $|\overrightarrow{r}-\overrightarrow{a}|=\pm \lambda |\overrightarrow{b}|$
From equation $\left(i\right)$
$\pm \lambda |\overrightarrow{b}|=6$
$\lambda =\pm \frac{6}{|\overrightarrow{b}|}$
Putting in eqution $\left(ii\right)$
$\overrightarrow{r}=\overrightarrow{a}\pm \frac{6}{|\overrightarrow{b}|}\overrightarrow{b}$