For real numbers α and β ≠ 0, if the point of intersection of the straight lines
(x - α)/1 = (y - 1)/2 = (z - 1)/3 and (x - 4)/β = (y - 6)/3 = (z - 7)/3, lies on the plane x + 2y - z = 8, then α - β is equal to :

Solution

Maths sir · 5 steps

Use the y and z equations first; they do not involve α or β, so they give the meeting point directly.

1. General points. A point on the first line is (α + t, 1 + 2t, 1 + 3t). A point on the second line is (4 + β s, 6 + 3s, 7 + 3s).
2. Match y and z. At the intersection, 1 + 2t = 6 + 3s and 1 + 3t = 7 + 3s. Subtracting the first from the second gives t = 1, and then 3 = 6 + 3s gives s = -1.
3. Use the plane. With t = 1 the point is (α + 1, 3, 4). Putting it in x + 2y - z = 8 gives α + 1 + 6 - 4 = 8, so α = 5 and the point is (6, 3, 4).
4. Match x. On the second line with s = -1, x = 4 - β. Setting 4 - β = 6 gives β = -2.
5. So. α - β = 5 - (-2) = 7, which is option (d).

The idea to keep. For two intersecting lines, first equate the coordinates that do not contain the unknown constants. That fixes the parameters, and the remaining equations then give the constants.

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