If the line of intersection of the planes a x + b y = 3 and a x + b y + c z = 0 , a > 0 makes an angle 30 ° with the plane y - z + 2 = 0 , then the direction cosines of the line are

Step-by-step solution

The vector along the line of intersection will be n → = i ^ j ^ k ^ a b 0 a b c = b c i ^ - a c j ^ i.e. direction ratios of line are b , - a , 0 Direction ratio of normal to the plane y - z + 2 = 0 are 0 , 1 , - 1 Since the line makes 30 ° with the plane, so it will make 60 ° with its normal. i.e. cos 60 ° = - a 2 b 2 + a 2 ⇒ a 2 a 2 + b 2 = 1 2 ⇒ b = ± a So, direction ratios of the line can be ± a , - a , 0 ∴ direction cosines can be ± ± 1 2 , - 1 2 , 0