The direction ratios of the line perpendicular to the lines $\frac{x=-5}{2}=\frac{y+11}{-3}=\frac{z+3}{1}$ and $\frac{x-7}{1}=\frac{y+2}{2}=\frac{z-4}{-2}$ are proportional to:

(4, 5, 7)

The correct answer is Option (4) → (4, 5, 7)

First line: $\frac{x-5}{2} = \frac{y+11}{-3} = \frac{z+3}{1}$

Direction ratios (d₁): $(2, -3, 1)$

Second line: $\frac{x-7}{1} = \frac{y+2}{2} = \frac{z-4}{-2}$

Direction ratios (d₂): $(1, 2, -2)$

Direction ratios of line perpendicular to both lines = cross product $(d₁ \times d₂)$

$d₁ \times d₂ = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 1 \\ 1 & 2 & -2 \end{vmatrix}$

$= \hat{i}[(-3)(-2) - (1)(2)] - \hat{j}[(2)(-2) - (1)(1)] + \hat{k}[(2)(2) - (-3)(1)]$

$= \hat{i}(6 - 2) - \hat{j}(-4 - 1) + \hat{k}(4 + 3)$

$= 4\hat{i} + 5\hat{j} + 7\hat{k}$

∴ Direction ratios are proportional to (4, 5, 7)

Final Answer:

(4, 5, 7)