List of top Mathematics Questions on Three Dimensional Geometry asked in JEE Main

If $(2\alpha + 1, \alpha^2 - 3\alpha, \frac{\alpha - 1}{2})$ is the image of $(\alpha, 2\alpha, 1)$ in the line $\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z}{1}$, then the possible value(s) of $\alpha$ is (are):

A line with direction ratios 1, -1, 2 intersects the lines $\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}$ and $\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}$ at the points P and Q, respectively. If the length of the line segment PQ is $\alpha$, then $225\alpha^2$ is equal to:

The square of the distance of the point (-2, -8, 6) from the line $\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{1}$ along the line $\frac{x+5}{1} = \frac{y+5}{1} = \frac{z}{2}$ is equal to:

Let the point $ A $ be the foot of perpendicular drawn from the point $ P(a, b, 0) $ on the line} \[ \frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - \alpha}{3}. \] If the midpoint of the line segment $ PA $ is $ \left( \frac{3}{4}, \frac{4}{3}, -\frac{1}{4} \right) $, then the value of $ a^2 + b^2 + \alpha^2 $ is:

Let the circles $ C_1 : |z| = r $ and $ C_2 : |z - 3 - 4i| = 5, z \in \mathbb{C} $, be such that $ C_2 $ lies within $ C_1 $. If $ z_1 $ moves on $ C_1 $, $ z_2 $ moves on $ C_2 $ and $ \min |z_1 - z_2| = 2 $, then $ \max |z_1 - z_2| $ is equal to:

Let the lines $L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 4\hat k)$, $\lambda \in \mathbb{R}$ and $L_2 : \vec r = (4\hat i + \hat j) + \mu(5\hat i + + 2\hat j + \hat k)$, $\mu \in \mathbb{R}$ intersect at the point $R$. Let $P$ and $Q$ be the points lying on lines $L_1$ and $L_2$, respectively, such that $|PR|=\sqrt{29}$ and $|PQ|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to}

If the distances of the point $ (1,2,a) $ from the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] along the lines \[ L_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} \quad \text{and} \quad L_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} \] are equal, then $ a+b+c $ is equal to:

If the image of the point $ P(3, 2, a) $ reflected about the line \[ \frac{x-3}{2} = \frac{y-5}{5} = \frac{z-2}{-2} \] is $ (5, b, c) $, then the value of $ \sigma^2 + b^2 + c^2 $ is:

The sum of all values of $\alpha$, for which the shortest distance between the lines $\dfrac{x+1}{\alpha}=\dfrac{y-2}{-1}=\dfrac{z-4}{-\alpha}$ and $\dfrac{x}{\alpha}=\dfrac{y-1}{2}=\dfrac{z-1}{2\alpha}$ is $\sqrt{2}$, is

Let the line $ L $ pass through the point $ (-3, 5, 2) $ and make equal angles with the positive coordinate axes. If the distance of $ L $ from the point $ (-2, r, 1) $ is $ \sqrt{\frac{14}{3}} $, then the sum of all possible values of $ r $ is :

Let $ L $ be the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} \] and let $ S $ be the set of all points $ (a,b,c) $ on $ L $, whose distance from the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} \] along the line $ L $ is $ 7 $. Then \[ \sum_{(a,b,c)\in S} (a+b+c) \] is equal to

The value of the integral $ \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} $ is :

Let $L$ be the distance of point $P(-1,2,5)$ from the line \[ \frac{x-1}{2}=\frac{y-3}{2}=\frac{z+1}{1} \] measured {parallel to a line having direction ratios $4,\,3,\,-5$. Then $L^2$ is equal to: }

If the image of a point $P(3,2,1)$ in the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] is $Q$, then the distance of $Q$ from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:

Let the line $L_1$ be parallel to the vector $-3\hat{i }+ 2\hat{j} + 4\hat{k}$ and pass through the point $(2, 6, 7)$, and the line $L_2$ be parallel to the vector $2\hat{i} + \hat{j} + 3\hat{k}$ and pass through the point $(4, 3, 5)$. If the line $L_3$ is parallel to the vector $-3\hat{i} + 5\hat{j} + 16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then $|\vec{CD}|^2$ is equal to :

If shortest distance between the lines \[ \frac{x+1}{\alpha}=\frac{y-2}{-2}=\frac{z-4}{-2\alpha} \quad \text{and} \quad \frac{x}{\alpha}=\frac{y-1}{1}=\frac{z-1}{\alpha} \] is $\sqrt{2}$, then find the sum of all possible values of $\alpha$.

Let a line $L$ passing through the point $P(1,1,1)$ be perpendicular to the lines \[ \frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1} \quad \text{and} \quad \frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}. \] Let the line $L$ intersect the $yz$-plane at the point $Q$.
Another line parallel to $L$ and passing through the point $S(1,0,-1)$ intersects the $yz$-plane at the point $R$.
Then the square of the area of the parallelogram $PQRS$ is equal to

If shortest distance between the lines \[ \frac{x+1}{\alpha}=\frac{y-2}{-2}=\frac{z-4}{-2\alpha} \quad \text{and} \quad \frac{x}{\alpha}=\frac{y-1}{1}=\frac{z-1}{\alpha} \] is $\sqrt{2}$, then find the sum of all possible values of $\alpha$.

If the foot is perpendicular from (1, 2, 3) to the line $\frac{x+1}{2} = \frac{y-2}{5} = \frac{z-1}{1}$ is $( a, \beta, \gamma)$, then find $a + \beta + \gamma$

If $A(1, 1, 1), B(0, λ, 0), C(λ + 1, 0, 1), D(2, -2, 2)$ are co-planer then $\sum (\lambda_i + 2)^2$ is equal to

$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$is equal to

The distance of the point $(-1,9,-16)$ from the plane $2 x+3 y-z=5$ measured parallel to the line $\frac{x+4}{3}=\frac{2-y}{4}=\frac{z-3}{12}$ is

The distance of the point $(7,-3,-4)$ from the plane passing through the points $(2,-3,1),(-1,1,-2)$ and $(3,-4,2)$ is :

Let the image of the point $P(2,-1,3)$ in the plane $x+2 y-z=0$ be $Q$ Then the distance of the plane $3 x+2 y+z+29=0$ from the point $Q$ is

The point of intersection $C$ of the plane $8 x+y+2 z=0$ and the line joining the points $A (-3,-6,1)$ and $B (2,4,-3)$divides the line segment $AB$ internally in the ratio$k : 1 \ If a , b , c (| a |,| b |, | c |$are coprime) are the direction ratios of the perpendicular from the point $C$on the line $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$, then $| a + b + c |$is equal to ___