$x + 1 = 2y = -12z, \quad x = y + 2 = 6z - 6$

$\frac{x + 1}{1} = \frac{y}{\frac{1}{2}} = \frac{z}{-\frac{1}{12}}, \quad \frac{x}{1} = \frac{y + 2}{1} = \frac{z - 1}{\frac{1}{6}}$

$\vec{r} = (-\hat{i} + 0\hat{j} + 0\hat{k}) + \lambda\left(\hat{i} + \frac{1}{2}\hat{j} - \frac{1}{12}\hat{k}\right)$

$\vec{r} = (0\hat{i} - 2\hat{j} + \hat{k}) + \beta\left(\hat{i} + \hat{j} + \frac{1}{6}\hat{k}\right)$

$d_1 = \left| \frac{(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2)}{|\vec{b}_1 \times \vec{b}_2|} \right| = \left| \frac{(\hat{i} - 2\hat{j} + \hat{k}) \cdot \left(\frac{1}{6}\hat{i} - \frac{1}{4}\hat{j} + \frac{1}{2}\hat{k}\right)}{\frac{7}{12}} \right|, \quad d_1 = \frac{\frac{1}{6} + \frac{1}{2} + \frac{1}{2}}{\frac{7}{12}} \implies d_1 = \frac{\frac{7}{6}}{\frac{7}{12}} = 2$

$\text{Now, } \frac{x - 1}{2} = \frac{y + 8}{-7} = \frac{z - 4}{5}, \quad \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 6}{-3}$

$\vec{a}_1 = \hat{i} - 8\hat{j} + 4\hat{k}, \quad \vec{a}_2 = \hat{i} + 2\hat{j} + 6\hat{k}$

$\vec{a}_2 - \vec{a}_1 = 0\hat{i} + 10\hat{j} + 2\hat{k}$

$\vec{b}_1 = 2\hat{i} - 7\hat{j} + 5\hat{k}, \quad \vec{b}_2 = 2\hat{i} + \hat{j} - 3\hat{k}$

$|\vec{b}_1 \times \vec{b}_2| = 16\sqrt{3}$

$(\vec{a}_2 - \vec{a}_1) \cdot (\vec{b}_1 \times \vec{b}_2) = 0 + 160 + 32 = 192$

$d_2 = \left| \frac{192}{16\sqrt{3}} \right| = \left| \frac{12}{\sqrt{3}} \right|$

$\therefore \frac{32\sqrt{3}d_1}{d_2} = \frac{32\sqrt{3} \times 2 \times \sqrt{3}}{12} = \frac{32 \times 6}{12} = 16$

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