A line in space is uniquely determined if(A) It passes through a give…

The correct answer is Option (2) → (A) and (B) only

Through a point with position vector \( \vec{a} \) and parallel to \( \vec{b} \) there is exactly one line:

    \( \displaystyle \vec{r}=\vec{a}+\lambda\vec{b} \)

Two distinct points \( \vec{a} \) and \( \vec{b} \) give the direction \( \vec{b}-\vec{a} \), so they also fix exactly one line:

    \( \displaystyle \vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a}) \)

So a line is fixed by a point together with a direction, or by two points; a point alone or a direction alone is not enough.

Checking the statements

(A) It passes through a given point and has a given direction. — Correct. The line is \( \vec{r}=\vec{a}+\lambda\vec{b} \).

(B) It passes through two given points. — Correct. The line is \( \vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a}) \).

(C) It passes through a given point. — Incorrect. Infinitely many lines pass through one point.

(D) It has a given direction. — Incorrect. Infinitely many parallel lines have the same direction.

NCERT (2026-27 edition):
Equation of a Line in Space