The line through the points A(1, −3, 4) and B(4, 3, 1) meets the line through the points C(−2, −3, 1) and D(4, 0, 4). The point of intersection is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (2, −1, 3)
The direction ratios of AB are 3, 6, −3, that is 1, 2, −1. The direction ratios of CD are 6, 3, 3, that is 2, 1, 1. The general points of the two lines are
\(AB:\ (1+\lambda,\ -3+2\lambda,\ 4-\lambda)\)
\(CD:\ (-2+2\mu,\ -3+\mu,\ 1+\mu)\)
Equating the y-coordinates:
\(-3+2\lambda = -3+\mu\), so \(\mu = 2\lambda\)
Equating the x-coordinates:
\(1+\lambda = -2+4\lambda\), so \(\lambda = 1,\ \mu = 2\)
The z-coordinates agree: \(4-1 = 3\) and \(1+2 = 3\). So the lines meet at \((1+1,\ -3+2,\ 3) = (2, -1, 3)\).
Why the other options are incorrect
Option (1) (3, 1, 2) — Incorrect. It lies on AB (\(\lambda = 2\)) but not on CD: \(-2+2\mu = 3\) gives \(\mu = \frac{5}{2}\), and then \(y = -\frac{1}{2} \ne 1\).
Option (2) (0, −2, 2) — Incorrect. It lies on CD (\(\mu = 1\)) but not on AB: \(1+\lambda = 0\) gives \(\lambda = -1\), and then \(y = -5 \ne -2\).
Option (3) (1, 1, 1) — Incorrect. On AB, x = 1 gives \(\lambda = 0\) and the point (1, −3, 4), not (1, 1, 1).