If the equation of the line passing through the point (0, −1/2, 0) and perpendicular to the lines r = λ(î + aĵ + bk̂) and r = ( î - ĵ - 6k̂) + μ(−bî + aĵ + 5k̂) is x-1/-2 = y+4/d = z-c/-4, then a + b + c + d is equal to :
❓ Question If the equation of the line passing through the point ( 0 , − 1 2 , 0 ) (0, -\dfrac{1}{2}, 0) and perpendicular to the lines r = λ ( i ^ + a j ^ + b k ^ ) and r = ( i ^ − j ^ − 6 k ^ ) + μ ( − b i ^ + a j ^ + 5 k ^ ) is x − 1 − 2 = y + 4 d = z − c − 4 , then a + b + c + d a + b + c + d is equal to: Question Image Short Solution We are given: A point P ( 0 , − 1 / 2 , 0 ). Two lines L 1 L_1 and L 2 L_2 . Required line is perpendicular to L 1 L_1 L 1 and L 2 L_2 L 2 and has equation x − 1 − 2 = y + 4 d = z − c − 4 \frac{x-1}{-2} = \frac{y+4}{d} = \frac{z-c}{-4} Steps: Direction ratios (DRs) of L 1 L_1 : 1 , a , b 1, a, b . DRs of L 2 L_2 : − b , a , 5 -b, a, 5 . The required line’s DRs are proportional to the cross product of these (since it is ⟂ to both). Use the point condition and given equation to find a , b , c , d a, b, c, d . Finally compute a + b + c + d a + b + c + d . Image Solution