Geometry

3D Geometry

Medium

core

The angle between the line $2x = 3y = z$ and $x$ - axis is:

$\cos^{-1}(\frac{1}{3})$

$\cos^{-1}(\frac{3}{7})$

$\cos^{-1}(\frac{2}{7})$

$\cos^{-1}(\frac{1}{7})$

✅ Correct Option: 2

The line is given as $2x = 3y = z$ .

This means all three expressions are equal to each other.

Let $2x = 3y = z = k$ where $k$ is a parameter.

From $2x = k$ :

$x = \frac{k}{2}$

From $3y = k$ :

$y = \frac{k}{3}$

From $z = k$ :

$z = k$

The direction ratios are $(\frac{1}{2}, \frac{1}{3}, 1)$ .

Multiplying by 6 to simplify:

Direction ratios = $(3, 2, 6)$

The $x$ -axis has direction ratios $(1, 0, 0)$ .

The angle $\theta$ between two lines is given by:

$\cos \theta = \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \times \sqrt{a_2^2 + b_2^2 + c_2^2}}$

where Line 1 has direction ratios $(3, 2, 6)$ and Line 2 has direction ratios $(1, 0, 0)$ .

The dot product:

$|3 \times 1 + 2 \times 0 + 6 \times 0| = |3| = 3$

For the line:

$\sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7$

For the $x$ -axis:

$\sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1$

$\cos \theta = \frac{3}{7 \times 1}$

$\cos \theta = \frac{3}{7}$

Therefore:

$\theta = \cos^{-1}(\frac{3}{7})$

2025: 3 June Shift 1

If the lines $\frac{x - 5}{5\lambda + 2} = \frac{2 - y}{5} = \frac{1 - z}{-1}$ and $x = \frac{y + 1/2}{2\lambda} = \frac{z - 1}{3}$ are perpendicular, then the value of $\lambda$ is equal to

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The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

2026: 30 May Shift 1

Let $L_1$ and $L_2$ lines such that $L_1:\vec{r}=\hat{i}+\hat{j}+\lambda(2\hat{i}-\hat{j}+\hat{k})$ , $L_2:\vec{r}=2\hat{i}+\hat{j}-\hat{k}+\mu(4\hat{i}-2\hat{j}+2\hat{k})$ and $\theta$ be the acute angle between $L_1$ and $L_2$ , then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\cos\theta$	I.	$\dfrac{\sqrt{66}}{6}$
B.	$\sin\theta$	II.	$3$
C.	the shortest distance (in units) between $L_1$ and $L_2$	III.	$1$
D.	(sum of squares of direction cosines of $L_1$ ) $+2$	IV.	$0$

Choose the correct answer from the options given below:

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If angle between the lines $\dfrac{x-1}{\propto} = \dfrac{y+2}{-5} = \dfrac{z+\frac{7}{3}}{\beta}$ and $x=z, y=0$ is $\dfrac{\pi}{4}$ , then

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The equation of line passing through the points (2, -1, 3) and (4, 2, 1) is

(A) $\frac{x-2}{2} = \frac{y+1}{3} = \frac{z-3}{-2}$

(B) $\frac{x-4}{3} = \frac{y-2}{-2} = \frac{z-1}{2}$

(D) $\frac{x-4}{-2} = \frac{y-2}{-3} = \frac{z-1}{2}$

Choose the correct answer from the options given below:

2026: 18 May Shift 2

If the two lines given by

$x=py+q, z=ry+s$ and

$x=p'y+q', z=r'y+s'$

are perpendicular, then which of the following is equal to -1 ?

The angle between the line 2x=3y=z2x = 3y = z2x=3y=z and xxx- axis is:

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The angle between the line $2x = 3y = z$ and $x$ - axis is: