Question

If \( \theta \) is the angle between the lines whose direction cosines are given by \( 6mn - 2nl + 5lm = 0 \) and \( 3l + m + 5n = 0 \), then \( \sin \theta = \)}

• A. \( \frac{\sqrt{35}}{6} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{\sqrt{37}}{6} \)
• D. \( \frac{5}{6} \)

Hint:

DRs calculation: substitute one linear equation into the quadratic one to find the ratio of variables.

Answer 1

The Correct Option is A

Step 1: Substitute l
From $3l+m+5n=0$, $m = -(3l+5n)$. Substitute into $6mn - 2nl + 5lm = 0$:
$-6n(3l+5n) - 2nl - 5l(3l+5n) = 0$.
Step 2: Solve for ratios
Simplifying gives $15l^2 + 45ln + 30n^2 = 0 \implies l^2 + 3ln + 2n^2 = 0$.
$(l+n)(l+2n) = 0 \implies l/n = -1$ or $l/n = -2$.
Step 3: Find DRs and \(\cos \theta\)
For $l_1:n_1 = -1 \implies l_1=-1, n_1=1, m_1=-2$.
For $l_2:n_2 = -2 \implies l_2=-2, n_2=1, m_2=1$.
$\cos \theta = \frac{|(-1)(-2) + (-2)(1) + (1)(1)|}{\sqrt{6}\sqrt{6}} = \frac{1}{6}$.
Step 4: Find \(\sin \theta\)
$\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - 1/36} = \frac{\sqrt{35}}{6}$.
Final Answer:(A)