Geometry

3D Geometry

Easy

core

P is a point on the line segment joining the points (3, 2, -1) and (6, 2, -2) internally. If x coordinate of P is 5, then its z-coordinate is

5/3

-5/3

-2

✅ Correct Option: 3

Parametrise as $(3+3t,\,2,\,-1-t)$ . Setting $3+3t=5$ gives $t=\frac23$ , so $z=-1-\frac23=-\frac53$ .

2026: 21 May Shift 2

Match the LIST-I with LIST-II

LIST-I (Line)		LIST-II (Direction cosiness)
A. $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(3\hat{i} + 2\hat{k})$		I. $\left\langle \dfrac{2}{\sqrt{5}}, \dfrac{1}{\sqrt{5}}, 0 \right\rangle$
B. $\dfrac{x-1}{2} = \dfrac{3-y}{1} = \dfrac{2z-1}{4}$		II. $\left\langle -\dfrac{1}{3}, \dfrac{2}{3}, -\dfrac{2}{3} \right\rangle$
C. $\dfrac{x+1}{1} = \dfrac{2y+1}{1}, z = -1$		III. $\left\langle \dfrac{3}{\sqrt{13}}, 0, \dfrac{2}{\sqrt{13}} \right\rangle$
D. $\dfrac{1-x}{1} = \dfrac{y}{2} = \dfrac{1-z}{2}$		IV. $\left\langle \dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3} \right\rangle$

Choose the correct answer from the options given below:

2025: 27 May Shift 1

Cosine of the acute angle between the lines $\frac{x-3}{2} = \frac{y-2}{1} = \frac{z-5}{2}$ and $\frac{x-1}{6} = \frac{y-3}{-3} = \frac{z+6}{2}$ is

2022: 4 Aug Shift 1

The equation of line of intersection of the planes $x + y + 3z = 7$ and $x - y + 2z = 3$ is:

2022: 10 Aug Shift 1

The distance of the plane $\vec{r} \cdot \left(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\right) = 2$ from the origin is

2026: 19 May Shift 1

If $\theta$ is the angle between the lines $L_1$ and $L_2$ , whose direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$ respectively, then

Match List-I with List-II

Choose the correct answer from the options given below:

2022: 25 May Shift 1

The angle between the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+5}{6}$ and the plane $2x + 10y - 11z = 5$ is:

Solution