Question

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` from the point (2, 1, 4) is `sqrt(3)`, is ______.

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Answer 1

The largest value of a, for which the perpendicular distance of the plane containing the lines `vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")`  from the point (2, 1, 4) is `sqrt(3)`, is 2.

Explanation:

Given vectors are

`vec"r" = (hat"i" + hat"j") + λ(hat"i" + "a"hat"j" - hat"k")` and `vec"r" = (hat"i" + hat"j") + μ(-hat"i" + hat"j" - "a"hat"k")` 

D.R’s of the plane containing these lines is

`|(hat"i", hat"j", hat"k"),(1, "a", -1),(-1, 1, -"a")| = hat"i"(1 - "a")^2 - hat"j"(-"a" - 1) + hat"k"(1 + "a")`

`vec"n" = (1 - "a")hat"i" + hat"j" + hat"k"`

One point in the plane : (1, 1, 0)

∴ equation of the plane is (1 – a)(x – 1) + (y – 1) + (z – 0) = 0

`\implies` (1 – a)x + y + z + a – 2 = 0

So,

D = `|(1 - "a")2 + 1 + 4 + "a" - 2|/sqrt((1 - "a")^2 + 1 + 1)`

`\implies` |5 – a| = `sqrt(3).sqrt("a"^2 - 2"a" + 3)`

`\implies` a² + 2a – 8 = 0

`\implies` a = 2, – 4

Therefore, largest value of a = 2