Let the point lie on the line of the shortest distance between the lines x + 2 – 3 = y – 2 4 = z – 5 2 and x + 2 – 1 = y + 6 2 = z – 1 0 . Then 2 is equal to _________. [2024]

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Solution

Q.
Let the point $(- 1, α, β)$ lie on the line of the shortest distance between the lines $\frac{x + 2}{- 3} = \frac{y - 2}{4} = \frac{z - 5}{2}$ and $\frac{x + 2}{- 1} = \frac{y + 6}{2} = \frac{z - 1}{0}$ . Then ${(α - β)}^{2}$ is equal to _________. [2024]

Ans.
(25)

Let $L_{1} : \frac{x + 2}{- 3} = \frac{y - 2}{4} = \frac{z - 5}{2}$ and $L_{2} : \frac{x + 2}{- 1} = \frac{y + 6}{2} = \frac{z - 1}{0}$

Let $P (- 3 λ - 2, 4 λ + 2, 2 λ + 5)$ be point on $L_{1}$ and $Q (- μ - 2, 2 μ - 6, 1)$ be point on $L_{2}$ .

Direction ratios of PQ = $(3 λ - μ, 2 μ - 4 λ - 8, - 2 λ - 4)$

Also, $\vec{P Q} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 3 & 4 & 2 \\ - 1 & 2 & 0 \end{matrix} |$ $= - 4 \hat{i} - 2 \hat{j} - 2 \hat{k}$

i.e., $2 \hat{i} + \hat{j} + \hat{k}$

$\Rightarrow \frac{3 λ - μ}{2} = \frac{2 μ - 4 λ - 8}{1} = \frac{- 2 λ - 4}{1}$

$\Rightarrow 3 λ - μ = - 4 λ - 8 and 2 μ - 4 λ - 8 = - 2 λ - 4$

$\Rightarrow μ = 7 λ + 8 and μ = λ + 2$

$\Rightarrow λ = - 1 and μ = 1$

$∴$ Equation of PQ is, $\frac{x + 3}{2} = \frac{y + 4}{1} = \frac{z - 1}{1}$

Now, $(- 1, α, β)$ lies on PQ

$\Rightarrow 1 = α + 4 = β - 1$

$\Rightarrow α = - 3, β = 2$

$∴ {(α - β)}^{2} = 25$ .

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