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(D) The formula for the cosine of the angle $	heta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is given by:$\cos

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$4$

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(D) The formula for the cosine of the angle $	heta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is given by:$\cos 	heta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$Given $	heta = 45^\circ$,so $\cos 45^\circ = \frac{1}{\sqrt{2}}$.Substituting the values:$\frac{1}{\sqrt{2}} = \frac{|2a - 3 + 10|}{\sqrt{2^2 + (-1)^2 + 2^2} \sqrt{a^2 + 3^2 + 5^2}}$$\frac{1}{\sqrt{2}} = \frac{|2a + 7|}{\sqrt{4 + 1 + 4} \sqrt{a^2 + 9 + 25}}$$\frac{1}{\sqrt{2}} = \frac{|2a + 7|}{3 \sqrt{a^2 + 34}}$Squaring both sides:$\frac{1}{2} = \frac{(2a + 7)^2}{9(a^2 + 34)}$$9(a^2 + 34) = 2(4a^2 + 28a + 49)$$9a^2 + 306 = 8a^2 + 56a + 98$$a^2 - 56a + 208 = 0$Solving the quadratic equation $(a - 52)(a - 4) = 0$,we get $a = 4$ or $a = 52$. Given the options,$a = 4$ is the correct value.

If the angle between the lines whose direction ratios are $2, -1, 2$ and $a, 3, 5$ is $45^\circ$,then $a =$

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