If $(2, - 1, 2)$ and $(K, 3, 5)$ are the triads of direction ratios of two lines and the angle between them is $45^{\circ}$ , then a value of $K$ is

2

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3

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4

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6

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Solution

The correct option is D $4$
Applying the formula for angle between 2 lines based on direction ratios :
$cos θ = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{3}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{3}}}$
$\frac{1}{\sqrt{2}} = \frac{2 K - 3 + 10}{3 \times \sqrt{K^{2} + 34}}$
On verification, we get $K = 4$

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