Three Dimensional Geometry — JEE Advanced PYQs

Let P be the plane such that it contains the straight line $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}-3}{3}=\frac{\mathrm{z}+2}{1}$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to $P$. T …

Let L be the straight line joining the points P(1,2, -1) and Q(2,3,1). Let S be the foot of the perpendicular drawn from the point $\mathrm{R}(4,-1,5)$ to the line L. Another line passing through R intersects L at a point T such that the point S divides the line segment PT intern …

Let $L_1$ be the line of intersection of the planes given by the equations

Let $L_2$ be the line passing through the point $P(2,-1,3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equation …

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1:x-y+3z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plan …

Let ${\mathbb{R}}^3$ denote the three-dimensional space. Take two points $\mathrm{P}=(1,2,3)$ and $\mathrm{Q}=(4,2,7)$.
Let dist (X, Y) denote the distance between two points X and Y in ${\mathbb{R}}^3$. Let …

Let $\gamma \in \mathbb{R}$ be such that the lines ${\mathrm{L}}_1:\frac{\mathrm{x}+11}{1}=\frac{\mathrm{y}+21}{2}=\frac{\mathrm{z}+29}{3}$ and ${\mathrm{L}}_2:\frac{\mathrm{x}+16}{3}=\frac{\mathrm{y}+11}{2}=\frac{\mathrm{z}+4}{\gamma }$ intersect. Let ${\mathrm{R}}_1$ b …

Let Q be the cube with the set of vertices $\left\{\left({\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\right)\in {\mathbb{R}}^3:{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3\{0,1\}\right\}$. Let F be the set of all twelve lines containing the diagonals of the six faces of the …

Let ${\ell }_1$ and ${\ell }_2$ be the lines ${\overrightarrow{\mathrm{r}}}_1=\lambda (\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})$ and ${\overrightarrow{\mathrm{r}}}_2=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu (\hat{\mathrm{i}}+\hat{\mathrm{k}})$, respectively. Let X be the se …

Let $P_1$ and $P_2$ be two planes given by


Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$ ?

Let $S$ be the reflection of a point $Q$ with respect to the plane given by

where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of …

Let $\alpha ,\beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma -1$
is consistent. Let $|M|$ represent the determinant of the matrix …

Let $L_1$ and $L_2$ be the following straight lines.
$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line
$L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$
lies in the plane containing $L_1$ and …

Let $\alpha ,\beta ,\gamma ,\delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ and $\alpha +\gamma =1$. Suppose the point $(3,2,-1)$ is the mirror image of the point $(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then wh …

Let ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ denotes the lines

respectively. If …

Three lines are given by

Let the lines cut the plane …

Three lines
$L_1:\vec{r}=\lambda \hat{i},\lambda \in \mathbb{R}$,
$L_2:\vec{r}=\vec{k}+\mu \hat{j},\mu \in \mathbb{R}$ and
$L_3:\vec{r}=\hat{i}+\hat{j}+v\hat{k},v\in \mathbb{R}$
are given. For which point(s) Q on $L_2$ can we find a point P on $L_1$ an …

Let $P_1:2x+y-z=3$ and $P_2:x+2y+z=2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the $z$-axis. Let the distance of $P$ from the $x$-a …

The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes $2x+y-2z=5$ and $3x-6y-2z=7$, is

Consider a pyramid $OPQRS$ located in the first octant $(x\ge 0,y\ge 0,z\ge 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP=3$. The point $S$ is directly above the mid-point …

Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

In ${\mathbb{R}}^3$, consider the planes $P_1:y=0$ and $P_2:x+z=1$. Let $P_3$ be a plane, different from $P_1$ and $P_2$, which passes through the intersection of $P_1$ and $P_2$. If the distance of the point $(0,1,0)$ from $P_3$ is 1 and the distance of a point …

In ${\mathbb{R}}^3$, let $L$ be a straight line passing through the origin. Suppose that all the points on $L$ are at a constant distance from the two planes $P_1:x+2y-z+1=0$ and $P_2:2x-y+z-1=0$. Let $M$ be the locus of the feet of the perpendiculars drawn from the point …

From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars $PQ$ and $PR$ are drawn respectively on the lines $y=x,z=1$ and $y=-x,z=-1$. If $P$ is such that $\angle QPR$ is a right angle, then the possible value(s) of $\lambda$ is(are)

Perpendiculars are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane $x+y+z=3$. The feet of perpendiculars lie on the line

A line $l$ passing through the origin is perpendicular to the lines

Then, the coordinate(s) of t …

Two lines $L_1:x=5,\frac{y}{3-\alpha }=\frac{z}{-2}$ and $L_2:x=\alpha ,\frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

Consider the lines $L_1:\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1},L_2:\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1:7x+y+2z=3,P_2:3x+5y-6z=4$. Let $ax+by+cz=d$ be the equation of the plane passing through the point of intersection of lines …

The equation of a plane passing through the line of intersection of the planes $x+2y+3z=2$ and $x-y+z=3$ and at a distance $\frac{2}{\sqrt{3}}$ from the point $(3,1,-1)$ is

If the straight lines $\frac{x-1}{2}=\frac{y+1}{k}=\frac{z}{2}$ and $\frac{x+1}{5}=\frac{y+1}{2}=\frac{z}{k}$ are coplanar, then the plane(s) containing these two lines is(are)

The point $P$ is the intersection of the straight line joining the points $Q(2,3,5)$ and $R(1,-1,4)$ with the plane $5x-4y-z=1$. If $S$ is the foot of the perpendicular drawn from the point $T(2,1,4)$ to $QR$, then the length of the line segment $PS$ is

If the distance of the point $\mathrm{P}(1,-2,1)$ from the plane $\mathrm{x}+2\mathrm{y}-2\mathrm{z}=\alpha$, where $\alpha >0$, is 5 , then the foot of the perpendicular from $P$ to the plane is

Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

If the distance between the plane $Ax-2y+z=d$ and the plane containing the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ is $\sqrt{6}$, then $|d|$ is

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane

at point $Q$. The length of the line segment $PQ$ equals

Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line

Then the value of $\mu$ for which the vector $\overrightarrow{PQ}$ is parallel to the plane $x-4y+3z=1$ is

Consider three planes

Let $L_1,L_2,L_3$ be the lines of intersection of the planes $P_2$ and $P_3,P_3$ and $P_1$, and $P_1$ and $P_2$, respectively.

STATEMENT …

Consider the lines


The shortest distance between $L_1$ and $L_2$ is

Consider the lines


The distance of the point $(1,1,1)$ from the plane passing through the point (-1, -2, -1) and whose normal is perpend …

Consider the planes $3x-6y-2z=15$ and $2x+y-2z=5$.

STATEMENT-1 : The parametric equations of the line of intersection of the given planes are $x=3+14t,y=1+2t,z=15t$.

because

STATEMENT-2 : The vector $14\hat{i}+2\hat{j}+15\hat{k}$ is parallel to the line of …

A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4$. The distance of the plane from the point (1, 2, 2) is