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Mathematics · JEE | THREE DIMENSIONAL GEOMETRY
हिन्दीIf the acute angle between the lines whose direction ratios are proportional to \( 2, 1, 2 \) and \( 4, -k, 3 \) is \( \cos^{-1}(\frac{2\sqrt{2}}{5}) \), then the value of \( k \) is:
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Correct answer · Option C
\( \frac{14}{17} \)
Explanation
Using \( \cos\theta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}} \), we have \( \frac{|8 - k + 6|}{3\sqrt{25+k^2}} = \frac{2\sqrt{2}}{5} \). Squaring both sides: \( 25(14-k)^2 = 72(25+k^2) \), which solves to give \( k = \frac{14}{17} \) (or negative root).
