Question

The point $\mathrm{P}$ is the intersection of the straight line joining the points $\mathrm{Q}(2,3,5)$ and $\mathrm{R}(1,-1,4)$with the plane $5x-4y-z=1$. If $\mathrm{S}$ is the foot of the perpendicular drawn from the point $\mathrm{T}(2,1,4)$ to $\mathrm{QR}$, then the length of the line segment $\mathrm{PS}$ is

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

Answer 1

The correct option is A

$\frac{1}{\sqrt{2}}$

Compute the required length:

Let's draw the figure.

The direction ratio of $\mathrm{QR}$ is $\left[(2-1),(3+1),(5-4)\right]$.

The direction ratio of $\mathrm{QR}$ is $\left[1,4,1\right]$.

The coordinate of $\mathrm{P}=\left(\frac{4}{3},\frac{1}{3},\frac{13}{3}\right)$.

The angle between $\mathrm{QR}$ and $\mathrm{PT}$ is $45°$,

$$\begin{gathered} \mathrm{PT}=1 \\ \mathrm{PS}=\mathrm{TS}=\frac{1}{\sqrt{2}} \end{gathered}$$

Hence option $\mathrm{A}$ is the correct option.