MHTCET 22 Apr 2025 Shift 1 Paper

Concept:For two parallel lines, the distance between them is the side length of the square.Perimeter of square $= 4 \times \text{side length}$.Explanation:The given lines have direction ratios $(2,3,4)$ for both, so they are parallel.A point on the first line is $\bar{r}_1 = (1,-2,3)$ and on the second line is $\bar{r}_2 = (0,1,-1)$.Thus, $\bar{r}_2 - \bar{r}_1 = (-1,3,-4)$.Distance between parallel lines is given by:$$D = \frac{\left|(\bar{r}_2 - \bar{r}_1) \times \bar{d}\right|}{|\bar{d}|}$$Here, $\bar{d} = (2,3,4)$.Compute the cross product:$$(\bar{r}_2 - \bar{r}_1) \times \bar{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 3 & -4 \\ 2 & 3 & 4 \end{vmatrix} = 24\hat{i} - 4\hat{j} - 9\hat{k}$$So, $\left|(\bar{r}_2 - \bar{r}_1) \times \bar{d}\right| = \sqrt{24^2 + (-4)^2 + (-9)^2} = \sqrt{673}$.Also, $|\bar{d}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{29}$.Therefore, side of the square:$$D = \frac{\sqrt{673}}{\sqrt{29}}$$Perimeter of the square:$$= 4D = \frac{4\sqrt{673}}{\sqrt{29}} \text{ units}$$Answer:$\frac{4\sqrt{673}}{\sqrt{29}}$ units, i.e., option B.