The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $$-$$ n = 0 and mn + nl + lm = 0, is :
JEE · Math · previous-year question
- A.$${\pi \over 2}$$correct
- B.$$\pi - {\cos ^{ - 1}}\left( {{4 \over 9}} \right)$$
- C.$${\cos ^{ - 1}}\left( {{8 \over 9}} \right)$$
- D.$${\pi \over 3}$$
Answer
A. $${\pi \over 2}$$
Explanation
n = 2 (l + m) lm + n(l + m) = 0 lm + 2(l + m)2 = 0 2l2 + 2m2 + 5ml = 0 $$2{\left( {{l \over m}} \right)^2} + 2 + 5\left( {{l \over m}} \right) = 0$$ 2t2 + 5t + 2 = 0 (t + 2)(2t + 1) = 0 $$ \Rightarrow t = - 2; - {1 \over 2}$$ (i) $${l \over m} = - 2$$ $${n \over m} = - 2$$ ($$-$$2m, m, $$-$$2m) ($$-$$2, 1, $$-$$2) (ii) $${l \over m} = - {1 \over 2}$$ n = $$-$$2l (l, $$-$$2l, $$-$$2l) (1, $$-$$2, $$-$$2) $$\cos \theta = {{ - 2 - 2 + 4} \over {\sqrt 9 \sqrt 9 }} = 0 \Rightarrow 0 = {\pi \over 2}$$
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