Three Dimensional Geometry — JEE Main PYQs

The point of intersection C of the plane $8x+y+2z=0$ 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 …

Shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$ and $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$ is

The plane $2\mathrm{x}-\mathrm{y}+\mathrm{z}=4$ intersects the line segment joining the points $\mathrm{A}(\mathrm{a},-2,4)$ and $\mathrm{B}(2,\mathrm{b},-3)$ at the point C in the ratio $2:1$ and the distance of the point C from the origin is $\sqrt{5}$. If …

If the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z+3}{1}$ and $\frac{x-a}{2}=\frac{y+2}{3}=\frac{z-3}{1}$ intersect at the point P, then the distance of the point P from the plane $z=a$ is:

Let the co-ordinates of one vertex of $△\mathrm{ABC}$ be $\mathrm{A}(0,2,\alpha )$ and the other two vertices lie on the line $\frac{x+\alpha }{5}=\frac{y-1}{2}=\frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $△\mathrm{ABC}$ is 21 sq. units …

Let the equation of the plane P containing the line $\mathrm{x}+10=\frac{8-\mathrm{y}}{2}=\mathrm{z}$ be $\mathrm{ax}+\mathrm{by}+3\mathrm{z}=2(\mathrm{a}+\mathrm{b})$ and the distance of the plane P from the point $(1,27,7)$ be c. Then …

The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is

The foot of perpendicular of the point $(2,0,5)$ on the line $\frac{x+1}{2}=\frac{y-1}{5}=\frac{z+1}{-1}$ is $(\alpha ,\beta ,\gamma )$. Then.
Which of the following is NOT correct?

If the shortest distance between the line joining the points $(1,2,3)$ and $(2,3,4)$, and the line $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-1}=\frac{\mathrm{z}-2}{0}$ is $\alpha$, then $28{\alpha }^2$ is equal to ______.

Let the plane containing the line of intersection of the planes
P1: $x+(\lambda +4)y+z=1$ and
P2: $2x+y+z=2$ pass through the points $(0,1,0)$
and $(1,0,1)$. Then the distance of the point
$(2\lambda ,\lambda ,-\lambda )$ from the plane P2 is

If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2y+z=0$ and $3y-z=3$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to

If the shortest between the lines
$\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and
$\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5}$ is 6, then the square of sum of all possible values of $\lambda$ is

Let P be the plane, passing through the point $(1,-1,-5)$ and perpendicular to the line joining the points $(4,1,-3)$ and $(2,4,3)$. Then the distance of P from the point $(3,-2,2)$ is

Let the plane P : $8\mathrm{x}+{\alpha }_1\mathrm{y}+{\alpha }_2\mathrm{z}+12=0$ be parallel to the line $\mathrm{L}:\frac{\mathrm{x}+2}{2}=\frac{\mathrm{y}-3}{3}=\frac{\mathrm{z}+4}{5}$. If the intercept of P on the y-axis is 1, then the distance between P and L is :

The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points $\mathrm{A},\mathrm{B},\mathrm{C}$ is $(2,\mathrm{a},4),\mathrm{a}\in \mathrm{N}$. If the volume of the tetrahedron OABC is 144 unit${}^3$, then which of the follow …

If a plane passes through the points $(-1,\mathrm{k},0),(2,\mathrm{k},-1)$, $(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2y+1}{2}=\frac{\mathrm{z}+1}{-1}$, then the value of $\frac{{\mathrm{k}}^2+1}{(\mathrm{k}-1)(\mathrm{k}-2)}$ is

Let a line L pass through the point P(2,3,1) and be parallel to the line $x+3y-2z-2=0=x-y+2z$. If the distance of L from the point $(5,3,8)$ is $\alpha$, then $3{\alpha }^2$ is equal to ____.

Let the shortest distance between the lines $L:\frac{x-5}{-2}=\frac{y-\lambda }{0}=\frac{z+\lambda }{1},\lambda \ge 0$ and $L_1:x+1=y-1=4-z$ be $2\sqrt{6}$. If $(\alpha ,\beta ,\gamma )$ lies on L, then which of the following is NOT possible?

Let $\theta$ be the angle between the planes $P_1=\vec{r}\cdot (\hat{i}+\hat{j}+2\hat{k})=9$ and $P_2=\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=15$. Let L be the line that meets ${\mathrm{P}}_2$ at the point ( $4,-2,5$ ) and makes an angle $\theta$ with the normal …

Let the line $L:\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2x+y+3z=16$ at the point $P$. Let the point Q be the foot of perpendicular from the point $\mathrm{R}(1,-1,-3)$ on the line L. If $\alpha$ is the area of triangle PQR, then ${\alpha }^2$ is equal t …

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

Consider the lines ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ given by
${\mathrm{L}}_1:\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-3}{1}=\frac{\mathrm{z}-2}{2}$
${\mathrm{L}}_2:\frac{\mathrm{x}-2}{1}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}-3}{3}$
A line ${\mathrm{L}}_3$ having direction …

The distance of the point $\mathrm{P}(4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to :

Let the equation of the plane passing through the line
$x-2y-z-5=0=x+y+3z-5$ and parallel to the line $\mathrm{x}+\mathrm{y}+2\mathrm{z}-7=0=2\mathrm{x}+3\mathrm{y}+\mathrm{z}-2$ be $\mathrm{ax}+\mathrm{by}+\mathrm{cz}=65$. Then the distance of the point …

Let Q be the foot of perpendicular drawn from the point $\mathrm{P}(1,2,3)$ to the plane $\mathrm{x}+2\mathrm{y}+\mathrm{z}=14$. If R is a point on the plane such that $\angle \mathrm{PRQ}=60^{\circ }$, then the area of $△\mathrm{PQR}$ is equal to:

The foot of the perpendicular from a point on the circle $x^2+y^2=1,z=0$ to the plane $2x+3y+z=6$ lies on which one of the following curves?

Let $\mathrm{P}(-2,-1,1)$ and $\mathrm{Q}\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha ,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, the …

Let the lines $\frac{x-1}{\lambda }=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{\mathrm{x}+26}{-2}=\frac{\mathrm{y}+18}{3}=\frac{\mathrm{z}+28}{\lambda }$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lies on P?

A plane P is parallel to two lines whose direction ratios are $-2,1,-3$, and $-1,2,-2$ and it contains the point $(2,2,-2)$. Let P intersect the co-ordinate axes at the points $\mathrm{A},\mathrm{B},\mathrm{C}$ making the intercepts $\alpha ,\beta ,\gamma$. If V is th …

If the plane P passes through the intersection of two mutually perpendicular planes $2x+ky-5z=1$ and $3kx-ky+z=5$, $k<3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane P on the y-axis is

Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and $4ax-y+5z-7a=0=2x-5y-z-3$, $a\in \mathbb{R}$ at the point $P(\alpha ,\beta ,\gamma )$. Then the value of $\alpha +\beta +\gamma$ equals …

Let P be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$ and $\frac{x}{3}=\frac{y}{7}=\frac{z}{8}$. If d is the distance of P from the point …

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and $\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of ${\sin }^{-1}\left(\sqrt{\frac{2}{27}}\right)$ with the plane $P:ax-y-z=0,(a>0)$. If the image of the point $(1,1,-5)$ in t …

The largest value of a, for which the perpendicular distance of the plane containing the lines $\vec{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+a\hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-a\hat{k})$ from the point $(2,1,4)$ is $\sqrt{3}$, is

The plane passing through the line $\mathrm{L}:\ell \mathrm{x}-\mathrm{y}+3(1-\ell )\mathrm{z}=1,x+2\mathrm{y}-\mathrm{z}=2$ and perpendicular to the plane $3x+2y+z=6$ is $3x-8y+7z=4$. If $\theta$ is the acute angle between the line L and the y-axis, then …

If the length of the perpendicular drawn from the point $P(a,4,2)$, $a>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}$ is $2\sqrt{6}$ units and $Q({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the image of the point P in this line, then $a+{\sum }_{i=1}^3{\alpha }_i$ is eq …

If the line of intersection of the planes $ax+by=3$ and $ax+by+cz=0$, $a>0$ makes an angle $30^{\circ }$ with the plane $y-z+2=0$, then the direction cosines of the line are :

A plane E is perpendicular to the two planes $2\mathrm{x}-2\mathrm{y}+\mathrm{z}=0$ and $\mathrm{x}-\mathrm{y}+2\mathrm{z}=4$, and passes through the point $\mathrm{P}(1,-1,1)$. If the distance of the plane E from the point $\mathrm{Q}(\mathrm{a},\mathrm{a},2)$ is …

The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is

The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0=x-2y+3z-5$ is :

Let Q and R be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then the square of the area of the triangle PQR is ______.

Let the plane $\mathrm{ax}+\mathrm{by}+\mathrm{cz}=\mathrm{d}$ pass through $(2,3,-5)$ and is perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$. If $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}$ are integers $\mathrm{d}>0$ and …

Let the image of the point $\mathrm{P}(1,2,3)$ in the line $\mathrm{L}:\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}-2}{3}$ be Q. Let $\mathrm{R}(\alpha ,\beta ,\gamma )$ be a point that divides internally the line segment PQ in the ratio $1:3$. Then the value o …

Let the foot of the perpendicular from the point $(1,2,4)$ on the line $\frac{x+2}{4}=\frac{y-1}{2}=\frac{z+1}{3}$ be $P$. Then the distance of P from the plane $3x+4y+12z+23=0$

The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$ is :

If two distinct point $\mathrm{Q},\mathrm{R}$ lie on the line of intersection of the planes $-x+2y-z=0$ and $3x-5y+2z=0$ and $PQ=PR=\sqrt{18}$ where the point P is $(1,-2,3)$, then the area of the triangle PQR is equal to

The acute angle between the planes ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$, when $P_1$ and $P_2$ are the planes passing through the intersection of the planes $5x+8y+13z-29=0$ and $8x-7y+z-20=0$ and the points $(2,1,3)$ and ( $0,1,2$ ), respectively, is

Let the plane $\mathrm{P}:\overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{a}}=\mathrm{d}$ contain the line of intersection of two planes $\overrightarrow{\mathrm{r}}\cdot (\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}})=6$ and …

Let $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}$ lie on the plane $px-qy+z=5$, for some $p,q\in \mathbb{R}$. The shortest distance of the plane from the origin is:

Let Q be the mirror image of the point $\mathrm{P}(1,2,1)$ with respect to the plane $x+2y+2z=16$. Let $T$ be a plane passing through the point Q and contains the line $\vec{r}=-\hat{k}+\lambda (\hat{i}+\hat{j}+2\hat{k}),\lambda \in \mathbb{R}$. Then, which of the following po …