If the equation of the line passing through the point ( 0, -1/2, 0 ) and perpendicular to the lines r⃗ = λ ( î + aĵ + bk̂ ) and r⃗ = ( î - ĵ - 6k̂ ) + μ ( -b î + aĵ + 5k̂ ) is (x-1)/-2 = (y+4)/d = (z-c)/-4 , then a+b+c+d is equal to :

Solution

Maths sir · 5 steps

Put the given point into the line first to get c and d, then perpendicularity gives a and b.

1. Point on line. The point (0, -1/2, 0) lies on (x-1)/-2 = (y+4)/d = (z-c)/-4. The x-part gives (0-1)/-2 = 1/2, so all three ratios equal 1/2.
2. Find d and c. From y: (-1/2+4)/d = 1/2, so 7/2 = d/2 and d = 7. From z: (0-c)/-4 = 1/2, so c = 2. The direction of the line is (-2, 7, -4).
3. Use perpendicularity. A line perpendicular to both given lines has a zero dot product with each direction vector, (1, a, b) and (-b, a, 5).

   -2 + 7a - 4b = 0, 2b + 7a - 20 = 0

4. Solve for a, b. Subtract the first equation from the second: 6b - 18 = 0, so b = 3. Then 7a = 2 + 12 = 14, so a = 2.
5. So. a + b + c + d = 2 + 3 + 2 + 7 = 14, which is option (b).

The idea to keep. When a line in symmetric form has unknowns and a known point on it, substitute the point first: the common ratio fixes the missing numbers. Perpendicular to two lines means zero dot product with both directions.

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