If the line (2-x)/3=(3 y-2)/(4 λ+1)=4-z makes a right angle with the line (x+3)/(3 μ)=(1-2 y)/6=(5-z)/7, then 4 λ+9 μ is equal to :

Solution

Maths sir · 4 steps

Rewrite both lines so x, y, z each have coefficient +1, then set the dot product of directions to zero.

1. First line. Make each numerator read (variable minus constant). (2-x)/3 = (x-2)/-3, (3y-2)/(4λ+1) = (y-2/3)/((4λ+1)/3) and 4-z = (z-4)/-1. So the direction is (-3, (4λ+1)/3, -1).
2. Second line. Do the same here. (1-2y)/6 = (y-1/2)/-3 and (5-z)/7 = (z-5)/-7. So the direction is (3μ, -3, -7).
3. Perpendicular lines. Two lines are at a right angle when the dot product of their direction ratios is zero.

   (-3)(3μ) + (4λ+1)/3(-3) + (-1)(-7) = 0 ⇒ -9μ - 4λ - 1 + 7 = 0

4. So. 4λ + 9μ = 6, which is option (d).

The idea to keep. Before reading direction ratios, put every term in the form (x-a)/l: flip signs and divide out any coefficient on x, y or z. Only then do the denominators give the direction.

Answer from the NTA final answer key. Solution by PrepWiser.