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Straight line in space — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsStraight line in space3 Marks
Question
It the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{2\lambda}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}-6}{-5}$ are perpendicular, find the value of $\lambda.$
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Answer

The diraction of ratios of the lines, $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{2\lambda}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}-6}{-5},$ are -3, 2k, 2 and 3k, 1, -5 respectiveiy.
It is know that two lines with direction ratios, a1, b1, c1 and a2, b2, c2, are perpendicular, if a1 a2 + b1 b2 + c1 c2 = 0
$\therefore$ -3 (3k) + 2k × 1 + 2 (-5) = 0
⇒ -9k + 2k - 10 = 0
⇒ 7k = - 10
$\Rightarrow\text{k}=\frac{-10}{7}$
Therefore, for $\text{k}=-\frac{10}{7},$ the given lines are perpendicular to each other.

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