Question

Let S be the set of all values of λ, for which the shortest distance between the lines x-λ/0 =y-3/4 = z+6/1 and x+λ/3 = y/-4 = z-6/0 is 13. Then 8 | $ \sum_{ λ∈S} λ| is $

• A. 302
• B. 304
• C. 306
• D. 308

Answer 1

The Correct Option is C

To solve the problem of finding the set of values of \( \lambda \) for which the shortest distance between the given lines is 13, we need to follow these steps:

1. Identify the direction ratios and points on each line:
   •  For the first line: \( \frac{x-\lambda}{0} = \frac{y-3}{4} = \frac{z+6}{1} \)
     • Direction ratios \( (a_1, b_1, c_1) = (0, 4, 1) \).
     • Point on the line: \((\lambda, 3, -6)\).
   • For the second line: \( \frac{x+\lambda}{3} = \frac{y}{-4} = \frac{z-6}{0} \)
     • Direction ratios \( (a_2, b_2, c_2) = (3, -4, 0) \).
     • Point on the line: \((- \lambda, 0, 6)\).
2. Use the formula for the shortest distance \( d \) between two skew lines:
3. Calculate \( \mathbf{n} = \mathbf{a}_1 \times \mathbf{a}_2 \):
4. Compute the vector joining any point on both lines:
5. Find the dot product \( \mathbf{r} \cdot \mathbf{n} \):
6. Substitute into the distance formula:
7. Solve the equation:
   • \(8\lambda + 135 = 169\)
   • \(8\lambda + 135 = -169\)

• \(8\lambda = 34 \Rightarrow \lambda = \frac{34}{8} = \frac{17}{4}\)
• \(8\lambda = -304 \Rightarrow \lambda = -38\)

1. Calculate \( \sum_{\lambda\in S} | \lambda |\):
2. Determine \(8 \mid \sum_{\lambda\in S} |\lambda| \):