If a line x+1 p = y-1 q = z-2 r where p = 2q = 3r , makes an angle with the positive direction of y -axis, then what is 2 equal to?

Step-by-step solution

The direction ratios of the line are proportional to p, q, r . Given p = 2q = 3r . Let p = 2q = 3r = 6k . Then p = 6k , q = 3k , r = 2k . The direction ratios of the line are proportional to 6, 3, 2 . The direction cosines of the line are 6 6^2+3^2+2^2 , 3 6^2+3^2+2^2 , 2 6^2+3^2+2^2 , which simplifies to 6 7 , 3 7 , 2 7 . The line makes an angle with the positive direction of the y -axis, so = 3 7 . Using the double angle formula, 2 = 2 ^2 - 1 . 2 = 2 ( 3 7 )^2 - 1 = 18 49 - 1 = - 31 49 . Answer: - 31 49