Question

The length of the perpendicular drawn from the points $(5,4,-1)$ to the line $\overline{r}=\hat{i}+\lambda \left(2\hat{i}+9\hat{i}+5\hat{k}\right)$ is

• A. $\frac{\sqrt{2190}}{110}$
• B. $\sqrt{\frac{2199}{110}}$
• C. $\sqrt{\frac{2109}{110}}$
• D. $\frac{\sqrt{23190}}{110}$

Answer 1

Answer: (C)

$\sqrt{\frac{2109}{110}}$

According to the question:

$\begin{array}{c}letthepoint(5,4,-1)bePandthepointthroughwhichthelinepassesbeQ(1,0,0).\\ thelineisparalleltothevector:\overrightarrow{r}=\left(2\hat{i}+9\hat{i}+5\hat{k}\right)\\ Now,\\ \overrightarrow{PQ}=-4\hat{i}-4\hat{j}+\hat{k}\\ \therefore \overrightarrow{r}\times \overrightarrow{PQ}=\left|\begin{array}{c}\hat{i}\hat{j}\hat{k}\\ 295\\ -4-41\end{array}\right|\\ =29\hat{i}-22\hat{j}+28\hat{k}\\ \Rightarrow \left|\overrightarrow{r}\times \overrightarrow{PQ}\right|=\sqrt{{\left(29\right)}^2+{(-22)}^2+{\left(28\right)}^2}\\ =\sqrt{841+484+784}=\sqrt{2109}\\ \left|\overrightarrow{r}\right|=\sqrt{2^2+9^2+5^2}\\ =\sqrt{4+81+25}=\sqrt{110}\\ d=\frac{\left|\overrightarrow{r}\times \overrightarrow{PQ}\right|}{\left|\overrightarrow{r}\right|}=\frac{\sqrt{2109}}{\sqrt{110}}=\sqrt{\frac{2109}{110}}\\ sothatthecorrectoptionisC.\end{array}$