Find the co-ordinates of the point on the line $\displaystyle \frac{x+2}{3}=\frac{\mathrm{y}+1}{2}=\frac{\mathrm{z}-3}{2}$ which is at a distance of $\displaystyle 5$ units from the point ($\displaystyle 1$, $\displaystyle 3$, $\displaystyle 3$).

Let the required point on given line be ( \(\displaystyle 3 \lambda-2,2 \lambda-1,2 \lambda+3\) ) for some \(\displaystyle \lambda\). According to question \[\begin{aligned} & \sqrt{(3 \lambda-3)^{2}+(2 \lambda-4)^{2}+(2 \lambda)^{2}}=5 \\ & 17 \lambda^{2}-34 \lambda+25=25 \\ & 17 \lambda(\lambda-2)=0 \text { gives } \lambda=0, \lambda=2 \end{aligned} \] ∴ Coordinates of required points are (-$\displaystyle 2$, -$\displaystyle 1$, $\displaystyle 3$), ($\displaystyle 4$, $\displaystyle 3$, $\displaystyle 7$)