Assertion (A) : The angle between the pair of lines
$$ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} \text{ and } \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} \text{ is } \cos^{-1}\left(\frac{8\sqrt{3}}{15}\right). $$
Reason (R) : The angle between the two lines is $\cos\theta = \left|\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}}\right|$ where $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are direction cosines of line $L_1$ and $L_2$ respectively.
In the light of the above statements, choose the most appropriate answer from the options given below :
Angle Between Lines MCQ Solution
Assertion (A) Verification
To find the angle between the two lines given in symmetric form, we first identify their direction ratios.
- Line 1 ($L_1$): $ \frac{x+3}{3} = \frac{y-1}{5} = \frac{z+3}{4} $ Direction ratios ($a_1, b_1, c_1$) are $(3, 5, 4)$.
- Line 2 ($L_2$): $ \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2} $ Direction ratios ($a_2, b_2, c_2$) are $(1, 1, 2)$.
The angle $\theta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is calculated using the formula:
$ \cos\theta = \left|\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}}\right| $Substitute the direction ratios:
- Numerator: $a_1a_2 + b_1b_2 + c_1c_2 = (3)(1) + (5)(1) + (4)(2) = 3 + 5 + 8 = 16$.
- Denominator part 1: $a_1^2 + b_1^2 + c_1^2 = 3^2 + 5^2 + 4^2 = 9 + 25 + 16 = 50$.
- Denominator part 2: $a_2^2 + b_2^2 + c_2^2 = 1^2 + 1^2 + 2^2 = 1 + 1 + 4 = 6$.
- $\cos\theta = \left|\frac{16}{\sqrt{50 \times 6}}\right| = \left|\frac{16}{\sqrt{300}}\right| = \left|\frac{16}{10\sqrt{3}}\right| = \frac{8}{5\sqrt{3}}$.
- Rationalizing the denominator gives: $\cos\theta = \frac{8\sqrt{3}}{15}$.
Thus, $\theta = \cos^{-1}\left(\frac{8\sqrt{3}}{15}\right)$. Assertion (A) is correct.
Reason (R) Analysis
Reason (R) provides the formula for the angle $\theta$ between two lines:
$ \cos\theta = \left|\frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}}\right| $Although it refers to $a_1, b_1, c_1$ etc. as 'direction cosines', the formula structure itself correctly uses direction ratios. This is the standard formula applied to find the angle between lines derived from their symmetric equations. Since Assertion (A) was verified using this exact formula structure, Reason (R) presents the correct method and is considered the correct explanation.
Conclusion
Both Assertion (A) and Reason (R) are correct. Reason (R) accurately states the formula used to calculate the angle between the lines, which validates Assertion (A). Therefore, Reason (R) is the correct explanation of Assertion (A).
The most appropriate answer is Option A.
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In the light of the above statements, choose the most appropriate answer from the options given below :
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