If the shortest distance between the lines \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{λ}\; and\; \frac{x-2}{1 }= \frac{y-4}{4 }= \frac{z-5}{5 }\;is\; \frac{1}{\sqrt3}\), then the sum of all possible values of \(λ\) is :
- 16
- 6
- 12
- 15
The Correct Option is A
Solution and Explanation
\(\overrightarrow a_1 = \hat i+2\hat j+3\hat k\)
\(\overrightarrow a_2 = 2\hat i + 4\hat j + 5\hat k\)
\(\overrightarrow p = 2\hat i+3\hat j+λ\hat k,\overrightarrow q = \hat i+4\hat j+5\hat k\)
\(∴ \overrightarrow p × \overrightarrow q = (15-4λ)\hat i-(10-λ)\hat j+5\hat k\)
∴ Shortest distance between the lines = \( | \frac{(15-4λ)-2(10-λ)+10}{√(15-4λ)^2+(10-λ)^2+25}| = \frac{1}{√3}\)
\(⇒ 5λ^2-80λ+275=0\)
the sum of all possible values of λ is : = \(\frac{80}{5} = 16\)
Hence, the correct option is (A): \(16\)
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Concepts Used:
Distance between Two Points
The distance between any two points is the length or distance of the line segment joining the points. There is only one line that is passing through two points. So, the distance between two points can be obtained by detecting the length of this line segment joining these two points. The distance between two points using the given coordinates can be obtained by applying the distance formula.
