Three Dimensional Geometry — JEE Main PYQs

If the plane $2x+y-5z=0$ is rotated about its line of intersection with the plane $3x-y+4z-7=0$ by an angle of $\frac{\pi }{2}$, then the plane after the rotation passes through the point :

If the lines $\overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda (3\hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}}=(\alpha \hat{\mathrm{i}}-\hat{\mathrm{j}})+\mu (2\hat{\mathrm{i}}-3\hat{\mathrm{k}})$ are co-planar, …

If the mirror image of the point $(2,4,7)$ in the plane $3x-y+4z=2$ is $(a,b,c)$, then $2a+b+2c$ is equal to :

Let d be the distance between the foot of perpendiculars of the points $\mathrm{P}(1,2,-1)$ and $\mathrm{Q}(2,-1,3)$ on the plane $-x+y+z=1$. Then $d^2$ is equal to

Let $P_1:\vec{r}\cdot (2\hat{i}+\hat{j}-3\hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points ( 2 , -3,2)(2,-2,-3) and (1,-4,2). If the direction ratios of the line of intersection of ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ be 16, …

If the two lines $l_1:\frac{x-2}{3}=\frac{y+1}{-2},z=2$ and $l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha }=\frac{z+5}{2}$ perpendicular, then an angle between the lines $l_2$ and $l_3:\frac{1-x}{3}=\frac{2y-1}{-4}=\frac{z}{4}$ is:

Let the plane $2x+3y+z+20=0$ be rotated through a right angle about its line of intersection with the plane $x-3y+5z=8$. If the mirror image of the point $\left(2,-\frac{1}{2},2\right)$ in the rotated plane is $B(a,b,c)$, then:

If two straight lines whose direction cosines are given by the relations $l+m-n=0,3l^2+m^2+cnl=0$ are parallel, then the positive value of c is :

Let the mirror image of the point $(a,b,c)$ with respect to the plane $3x-4y+12z+19=0$ be $(a-6,\beta ,\gamma )$. If $a+b+c=5$, then $7\beta -9\gamma$ is equal to ______.

If the foot of the perpendicular from the point $\mathrm{A}(-1,4,3)$ on the plane $\mathrm{P}:2\mathrm{x}+\mathrm{my}+\mathrm{nz}=4$, is $\left(-2,\frac{7}{2},\frac{3}{2}\right)$, then the distance of the point A from the plane P, measured parallel to a line with directi …

Let a line with direction ratios a, $-4\mathrm{a},-7$ be perpendicular to the lines with direction ratios 3, $-1,2b$ and $b,a,-2$. If the point of intersection of the line $\frac{x+1}{a^2+b^2}=\frac{y-2}{a^2-b^2}=\frac{z}{1}$ and the plane …

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the point A and B. Then $(\mathrm{AB}{)}^2$ is equal to ______.

If the shortest distance between the line
$\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+3\mathrm{k})+\lambda (\hat{\mathrm{i}}-\mathrm{a}\hat{\mathrm{j}})$ and
$\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{j}}+2\mathrm{k})+\mu (\hat{\mathrm{i}}-\hat{\mathrm{j}}+\mathrm{k})$ …

If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda }$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$, then the sum of all possible values of $\lambda$ is :

Let the points on the plane P be equidistant from the points $(-4,2,1)$ and $(2,-2,3)$. Then the acute angle between the plane P and the plane $2x+y+3z=1$ is

Let P be the plane passing through the intersection of the planes
$\vec{r}\cdot (\hat{i}+3\hat{j}-\hat{k})=5$ and $\vec{r}\cdot (2\hat{i}-\hat{j}+\hat{k})=3$, and the point $(2,1,-2)$. Let the position vectors of the points $X$ and $Y$ be $\hat{i}-2\hat{j}+4\hat{k}$ and …

Let $l_1$ be the line in xy-plane with x and y intercepts $\frac{1}{8}$ and $\frac{1}{4\sqrt{2}}$ respectively, and $l_2$ be the line in zx-plane with x and z intercepts $-\frac{1}{8}$ and $-\frac{1}{6\sqrt{3}}$ respectively. If d is the shortest distance between the line …

Let Q be the mirror image of the point $\mathrm{P}(1,0,1)$ with respect to the plane $\mathrm{S}:\mathrm{x}+\mathrm{y}+\mathrm{z}=5$. If a line L passing through $(1,-1,-1)$, parallel to the line PQ meets the plane S at R, then ${\mathrm{QR}}^2$ is equal to:

Let the lines
${\mathrm{L}}_1:\overrightarrow{\mathrm{r}}=\lambda (\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}),\lambda \in \mathrm{R}$
$L_2:\vec{r}=(\hat{i}+3\hat{j}+\hat{k})+\mu (\hat{i}+\hat{j}+5\hat{k});\mu \in R$
intersect at the point S. If a plane …

A plane $P$ contains the line $x+2y+3z+1=0=x-y-z-6$, and is perpendicular to the plane $-2x+y+z+8=0$. Then which of the following points lies on $P$ ?

Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2y-z=3$. If $d$ is the distance of the point $(0,0,6)$ from $L$, then $d^2$ is equal to ____.

The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane
$x+y+z=17$ is:

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

Let P be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+4\hat{k})=16$ and $\vec{r}\cdot (-\hat{i}+\hat{j}+\hat{k})=6$.
Then which of the following points does NOT lie on P ?

A hall has a square floor of dimension $10\mathrm{m}\times 10\mathrm{m}$ (see the figure) and vertical walls. If the angle GPH between diagonals AG and BH is ${\cos }^{-1}\frac{1}{5}$, then the height of the hall (in meters) is:

Let Q be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and $\frac{x-1}{3}=\frac{y-2}{5}=\frac{z-3}{7}$. Then $(PQ{)}^2$ is equal to ____.

If the shortest distance between the lines ${\vec{r}}_1=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k}),\lambda \in R$, $\alpha >0$ and ${\vec{r}}_2=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k}),\mu \in R$ is 9, then $\alpha$ is equal to _ …

Let $\mathrm{a},\mathrm{b}\in \mathrm{R}$. If the mirror image of the point $\mathrm{P}(\mathrm{a},6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is
$(20,\mathrm{b},-\mathrm{a}-9)$, then $|\mathrm{a}+\mathrm{b}|$ is equal to:

The vector equation of the plane passing through the intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r}\cdot (\hat{i}-2\hat{j})=-2$, and the point $(1,0,2)$ is:

Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda =2y-1=-2z$ and $x=y+2\lambda =z-\lambda$ is $\frac{\sqrt{7}}{2\sqrt{2}}$, then the value of $|\lambda |$ is

Consider the line L given by the equation $\frac{\mathrm{x}-3}{2}=\frac{\mathrm{y}-1}{1}=\frac{\mathrm{z}-2}{1}$. Let Q be the mirror image of the point $(2,3,-1)$ with respect to $L$. Let a plane $P$ be such that it passes through $Q$, and the line L is perpendicular t …

The lines $x=ay-1=z-2$ and $x=3y-2=bz-2,(ab\ne 0)$ are coplanar, if :

The angle between the straight lines, whose direction cosines are given by the equations $2\ell +2\mathrm{m}-\mathrm{n}=0$ and $\mathrm{mn}+\mathrm{nI}+\mathrm{Im}=0$, is:

The equation of the plane passing through the line of intersection of the planes $\vec{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r}\cdot (2\hat{i}+3\hat{j}-\hat{k})+4=0$ and parallel to the $x-$ axis is

Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x-y+z+3=0$ and let $\mathrm{R}(3,5,\gamma )$ be a point of this plane. Then the square of the length of the line segment SR is

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^2+m^2-n^2=0$. Then the value of ${\sin }^4\alpha +{\cos }^4\alpha$ is :

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :

If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda )=(z-3),\lambda \in R$ is $\frac{1}{\sqrt{38}}$, then the integral value of $\lambda$ is equal to:

Let $L$ be the line of intersection of planes $\vec{r}\cdot (\hat{i}-\hat{j}+2\hat{k})=2$ and $\vec{r}\cdot (2\hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular on $L$ from the point $(1,2,0)$, then the value of …

If $(1,5,35),(7,5,5),(1,\lambda ,7)$ and $(2\lambda ,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

Consider the three planes
$P_1:3x+15y+21z=9$,
$P_2:x-3y-z=5$, and
$P_3:2x+10y+14z=5$
Then, which one of the following is true ?

Let $(\lambda ,2,1)$ be a point on the plane which passes which through the point $(4,-2,2)$. If the plane is perpendicular to the line joining the points $(-2,-21,29)$ and $(-1,-16,23)$, then ${\left(\frac{\lambda }{11}\right)}^2-\frac{4\lambda }{11}-4$ is equal to ______

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to a line, whose direction ratios are $2,3,-6$ is

Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2x+y-3z+4=0$, is :

If the mirror image of the point ( $1,3,5$ ) with respect to the plane $4x-5y+2z=8$ is ( $\alpha ,\beta ,\gamma$ ) then $5(\alpha +\beta +\gamma )$ equals :

Let $L$ be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from $(3,2,1)$ on $L$, then the value of $21(\alpha +\beta +\gamma )$ equals :

The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2x+3y+2z=0$ and $x-2y+z=0$ is :

Suppose the line $\frac{x-2}{\alpha }=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3y-2z+\beta =0$. Then $(\alpha +\beta )$ is equal to ____.

Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3x-2y+4z-7=0$ and $x+5y-2z+9=0$, be $\alpha x+\beta y+\gamma z+3=0$, then $\alpha +\beta +\gamma$ is equal to:

The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2x-y+z=6$ from the point $(-1,-1,2)$ is ____.