The Cartesian equation of the line passing through the point (2, −1, 3) and making equal acute angles with the coordinate axes is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{x-2}{1}=\frac{y+1}{1}=\frac{z-3}{1}\)
If the line makes the same angle \(\alpha\) with each axis, its direction cosines are all equal:
\(l = m = n = \cos\alpha\)
The angle is acute, so \(\cos\alpha \gt 0\) and the direction ratios can be taken as 1, 1, 1.
The line through (2, −1, 3) with direction ratios 1, 1, 1 is
\( \displaystyle \frac{x-2}{1}=\frac{y+1}{1}=\frac{z-3}{1} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{x}{2}=\frac{y}{-1}=\frac{z}{3}\) — Incorrect. This is the line through the origin with direction ratios 2, −1, 3; these are not equal.
Option (2) \(\displaystyle\frac{x+2}{1}=\frac{y-1}{1}=\frac{z+3}{1}\) — Incorrect. This passes through (−2, 1, −3), not (2, −1, 3).
Option (4) \(\displaystyle\frac{x-2}{1}=\frac{y+1}{1}=\frac{z-3}{-1}\) — Incorrect. With direction ratios 1, 1, −1 the line makes an obtuse angle with the z-axis while the other two are acute.