A line 'l' passing through origin is perpendicular to the lines
l₁:r⃗ = (3 + t)î + ( - 1 + 2t)ĵ + (4 + 2t)k̂
l₂:r⃗ = (3 + 2s)î + (3 + 2s)ĵ + (2 + s)k̂
If the co-ordinates of the point in the first octant on 'l₂‘ at a distance of √17 from the point of intersection of 'l' and 'l₁' are (a, b, c) then 18(a + b + c) is equal to ___________.

Solution

Maths sir · 5 steps

Cross the two directions to get l, meet it with l₁, then walk along l₂ a distance √17.

1. Direction of l. The directions are (1, 2, 2) for l₁ and (2, 2, 1) for l₂. A line perpendicular to both runs along their cross product, (1,2,2) × (2,2,1) = (-2, 3, -2). Since l passes through the origin, its points are λ(-2, 3, -2).
2. Meet l1. Equate the x and z coordinates: 3 + t = -2λ and 4 + 2t = -2λ, so t = -1 and λ = -1. The y coordinate checks out (-1 + 2t = -3 = 3λ), so the point of intersection is P(2, -3, 2).
3. Distance on l2. A general point on l₂ is (3 + 2s, 3 + 2s, 2 + s). Its distance from P must be √17, so the squared distance is 17.

   (1+2s)² + (6+2s)² + s² = 17 ⇒ 9s² + 28s + 20 = 0 ⇒ (9s+10)(s+2) = 0

4. Pick the root. s = -2 gives (-1, -1, 0), which is not in the first octant. s = -10/9 gives (7/9, 7/9, 8/9), with every coordinate positive.
5. So. a + b + c = 22/9, and 18 × 22/9 = 44.

The common slip: -36. Using the root s = -2 gives the point (-1, -1, 0) and 18 × (-2) = -36, but that point is not in the first octant.

The idea to keep. A line perpendicular to two lines runs along the cross product of their directions. For a point at a given distance on a line, the general point gives a quadratic, and the extra condition picks the root.

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