Three Dimensional Geometry — JEE Main PYQs

Let the line of the shortest distance between the lines
$L_1:\vec{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and
$L_2:\vec{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (\hat{i}+\hat{j}-\hat{k})$
intersect ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ at P a …

Let PQR be a triangle with $\mathrm{R}(-1,4,2)$. Suppose $\mathrm{M}(2,1,2)$ is the mid point of PQ. The distance of the centroid of $△\mathrm{PQR}$ from the point of intersection of the line $\frac{\mathrm{x}-2}{0}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}+3}{-1}$ and …

A line with direction ratios $2,1,2$ meets the lines $x=y+2=z$ and $x+2=2y=2z$ respectively at the point P and Q. If the length of the perpendicular from the point $(1,2,12)$ to the line PQ is $l$, then $l^2$ is

Let $L_1:\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}-\hat{j}+2\hat{k}),\lambda \in R$
$L_2:\vec{r}=(\hat{j}-\hat{k})+\mu (3\hat{i}+\hat{j}+p\hat{k}),\mu \in R$ and …

Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1:\frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $\mathrm{M}(\alpha ,\beta ,\gamma )$ and ${\mathrm{L}}_2:\frac{\mathrm{x}+2}{-3}=\frac{\mathrm{y}-2}{-2}=\frac{\mathrm{z}-1}{4}$ at …

Let the line L pass through the point $(0,1,2)$, intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$. Then the distance of the point $P(1,-9,2)$ from the line L is

A plane P contains the line of intersection of the plane $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=6$ and $\vec{r}\cdot (2\hat{i}+3\hat{j}+4\hat{k})=-5$. If P passes through the point $(0,2,-2)$, then the square of distance of the point (12, 12, 18) from the plane P is

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha }$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta }$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is

If the equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda (\hat{i}+\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k})$ is $\mathrm{ax}+\mathrm{by}+\mathrm{cz}=4$, then …

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is

Let ${\lambda }_1,{\lambda }_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2},1,\lambda \right)$ and $(-2,0,1)$ are at equal distance from the plane $2x+3y-6z+7=0$. If ${\lambda }_1>{\lambda }_2$, then the distance of the point …

If the equation of the plane passing through the line of intersection of the planes $2\mathrm{x}-\mathrm{y}+\mathrm{z}=3,4\mathrm{x}-3\mathrm{y}+5z+9=0$ and parallel to the line $\frac{\mathrm{x}+1}{-2}=\frac{\mathrm{y}+3}{4}=\frac{\mathrm{z}-2}{5}$ is …

One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x , y and z axes are 3, 4 and 5 units respectively. Let P be the vertex $(3,4,5)$. Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing thr …

Let the image of the point $P(1,2,3)$ in the plane $2x-y+z=9$ be $Q$. If the coordinates of the point $R$ are $(6,10,7)$, then the square of the area of the triangle PQR is

Let P be the plane passing through the line $\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$ and the point $(2,4,-3)$. If the image of the point $(-1,3,4)$ in the plane P is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to

For $\mathrm{a},\mathrm{b}\in \mathrm{Z}$ and $|\mathrm{a}-\mathrm{b}|\le 10$, let the angle between the plane $P$ : $ax+y-z=b$ and the line $l:\mathrm{x}-1=\mathrm{a}-\mathrm{y}=\mathrm{z}+1$ be ${\cos }^{-1}\left(\frac{1}{3}\right)$. If the distance of the point …

Let $P_1$ be the plane $3x-y-7z=11$ and $P_2$ be the plane passing through the points $(2,-1,0)$, $(2,0,-1)$, and $(5,1,1)$. If the foot of the perpendicular drawn from the point $(7,4,-1)$ on the line of intersection of the planes $P_1$ and $P_2$ is …

Let the image of the point $\mathrm{P}(1,2,6)$ in the plane passing through the points $\mathrm{A}(1,2,0),\mathrm{B}(1,4,1)$ and $\mathrm{C}(0,5,1)$ be $\mathrm{Q}(\alpha ,\beta ,\gamma )$. Then $\left({\alpha }^2+{\beta }^2+{\gamma }^2\right)$ is equal to :

Let the line $\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plan …

Let the foot of perpendicular from the point $\mathrm{A}(4,3,1)$ on the plane $P:x-y+2z+3=0$ be $N$. If $B(5,\alpha ,\beta ),\alpha ,\beta \in \mathbb{Z}$ is a point on plane P such that the area of the triangle ABN in $3\sqrt{2}$, then …

Let P be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane $x+y+z=2$.

If the distance of the point P from the plane $3x-4y+12z=32$ is $q$, then $q$ and $2q$ are the roots of the equation

Let two vertices of triangle ABC be $(2,4,6)$ and $(0,-2,-5)$, and its centroid be $(2,1,-1)$. If the image of third vertex in the plane $x+2y+4z=11$ is $(\alpha ,\beta ,\gamma )$, then $\alpha \beta +\beta \gamma +\gamma \alpha$ is equal to

The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is

Let the lines $l_1:\frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha }{-2}$ and $l_2:3x+2y+z-2=0=x-3y+2z-13$ be coplanar. If the point $P(a,b,c)$ on $l_1$ is nearest to the point $Q(-4,-3,2)$, then $|a|+|b|+|c|$ is equal to

Let the plane $\mathrm{P}:4\mathrm{x}-\mathrm{y}+\mathrm{z}=10$ be rotated by an angle $\frac{\pi }{2}$ about its line of intersection with the plane $\mathrm{x}+\mathrm{y}-\mathrm{z}=4$. If $\alpha$ is the distance of the point $(2,3,-4)$ from the new position of the …

Let the plane $\mathrm{x}+3\mathrm{y}-2\mathrm{z}+6=0$ meet the co-ordinate axes at the points $\mathrm{A},\mathrm{B},\mathrm{C}$. If the orthocentre of the triangle ABC is $\left(\alpha ,\beta ,\frac{6}{7}\right)$, then $98(\alpha +\beta {)}^2$ is equal to

The plane, passing through the points $(0,-1,2)$ and $(-1,2,1)$ and parallel to the line passing through $(5,1,-7)$ and $(1,-1,-1)$, also passes through the point

Let N be the foot of perpendicular from the point P $(1,-2,3)$ on the line passing through the points $(4,5,8)$ and $(1,-7,5)$. Then the distance of N from the plane $2x-2y+z+5=0$ is

The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, is

The distance of the point $(-1,2,3)$ from the plane $\overrightarrow{\mathrm{r}}.(\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}})=10$ parallel to the line of the shortest distance between the lines $\vec{r}=(\hat{i}-\hat{j})+\lambda (2\hat{i}+\hat{k})$ and …

Let the equation of plane passing through the line of intersection of the planes $x+2y+az=2$ and $x-y+z=3$ be $5x-11y+bz=6a-1$. For $c\in \mathbb{Z}$, if the distance of this plane from the point $(a,-c,c)$ is $\frac{2}{\sqrt{a}}$, then $\frac{a+b}{c}$ is equal to

Let the image of the point $\left(\frac{5}{3},\frac{5}{3},\frac{8}{3}\right)$ in the plane $x-2y+z-2=0$ be $P$. If the distance of the point $Q(6,-2,\alpha )$, $\alpha >0$, from $P$ is 13, then $\alpha$ is equal to _____.

Let S be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{\mathrm{x}-\lambda }{0}=\frac{\mathrm{y}-3}{4}=\frac{\mathrm{z}+6}{1}$ and $\frac{\mathrm{x}+\lambda }{3}=\frac{\mathrm{y}}{-4}=\frac{\mathrm{z}-6}{0}$ is 13. Then …

Let the foot of perpendicular of the point $\mathrm{P}(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $\mathrm{Q}(\alpha ,\beta ,\gamma )$. Then the distance of Q from the origin is:

Let the plane P contain the line
$2\mathrm{x}+\mathrm{y}-\mathrm{z}-3=0=5\mathrm{x}-3\mathrm{y}+4\mathrm{z}+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point $\mathrm{A}(8,-1,-19)$ from the plane P measured paral …

Let the line passing through the points, $\mathrm{P}(2,-1,2)$ and $\mathrm{Q}(5,3,4)$ meet the plane $\mathrm{x}-\mathrm{y}+\mathrm{z}=4$ at the point R. Then the distance of the point R from the plane $\mathrm{x}+2\mathrm{y}+3\mathrm{z}+2=0$ measured parallel to the li …

Let P be the plane passing through the points (5, 3, 0), (13, 3, -2) and (1, 6, 2). For $\alpha \in \mathrm{N}$, if the distances of the points $\mathrm{A}(3,4,\alpha )$ and $\mathrm{B}(2,\alpha ,a)$ from the plane P are 2 and 3 respectively, then the positive value of …

Let the line $\ell :\mathrm{x}=\frac{1-\mathrm{y}}{-2}=\frac{\mathrm{z}-3}{\lambda },\lambda \in \mathbb{R}$ meet the plane $\mathrm{P}:\mathrm{x}+2\mathrm{y}+3\mathrm{z}=4$ at the point $(\alpha ,\beta ,\gamma )$. If the angle between the line $\ell$ and the plane P is …

If equation of the plane that contains the point $(-2,3,5)$ and is perpendicular to each of the planes $2x+4y+5z=8$ and $3x-2y+3z=5$ is $\alpha x+\beta y+\gamma z+97=0$ then $\alpha +\beta +\gamma =$

Let $(\alpha ,\beta ,\gamma )$ be the image of the point $\mathrm{P}(2,3,5)$ in the plane $2\mathrm{x}+\mathrm{y}-3\mathrm{z}=6$. Then $\alpha +\beta +\gamma$ is equal to

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector …

Let a line $l$ pass through the origin and be perpendicular to the lines
$l_1:\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-11\hat{\mathrm{j}}-7\hat{\mathrm{k}})+\lambda (\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}),\lambda \in \mathbb{R}$
and …

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 :

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

The shortest distance between the lines $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-6}{2}$ and $\frac{x-6}{3}=\frac{1-y}{2}=\frac{z+8}{0}$ is equal to

Let a unit vector $\overrightarrow{\mathrm{OP}}$ make angle $\alpha ,\beta ,\gamma$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\beta \in \left(0,\frac{\pi }{2}\right)$. $\overrightarrow{\mathrm{OP}}$ is perpendicular to the plane t …

If ${\lambda }_1$ and ${\lambda }_2$ are two values of $\lambda$ such that the angle between the planes $P_1:\vec{r}\cdot (3\hat{i}-5\hat{j}+\hat{k})=7$ and $P_2:\vec{r}\cdot (\lambda \hat{i}+\hat{j}-3\hat{k})=9$ is ${\sin }^{-1}\left(\frac{2\sqrt{6}}{5}\right)$, the …

If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3y+2z-1=0$, $4x-y+z=0$ is $Ax+By+Cz=1$, then $140(C-B+A)$ is equal to ____.

Let the plane P pass through the intersection of the planes $2x+3y-z=2$ and $x+2y+3z=6$, and be perpendicular to the plane $2\mathrm{x}+\mathrm{y}-\mathrm{z}+1=0$. If d is the distance of P from the point $(-7,1,1)$, then ${\mathrm{d}}^2$ is equal to :

Let $\alpha x+\beta y+\gamma z=1$ be the equation of a plane passing through the point $(3,-2,5)$ and perpendicular to the line joining the points $(1,2,3)$ and $(-2,3,5)$. Then the value of $\alpha \beta \gamma$ is equal to ______.