3D Geometry · JEE Main previous year questions

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 :

1. A.3
2. B.1
3. C.$-1$
4. D.5

Let the plane containing the line of intersection of the planes ${\mathrm{P}}_1:x+(\lambda +4)y+z=1$ and ${\mathrm{P}}_2:2x+y+z=2$ pass through the points $(0,1,0)$ and $(1,0,1)$. Then the distance of the point $(2\lambda ,\lambda ,-\lambda )$ from the plane ${\mathrm{P}}_2$ is :

1. A.$2\sqrt{6}$
2. B.$3\sqrt{6}$
3. C.$4\sqrt{6}$
4. D.$5\sqrt{6}$

If the shortest distance 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 :

If the foot of the perpendicular from the point $\mathrm{A}(-1,4,3)$ on the plane $\mathrm{P}:2x+my+nz=4$, 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 direction ratios $3,-1,-4$, is equal to :

1. A.1
2. B.$\sqrt{26}$
3. C.$2\sqrt{2}$
4. D.$\sqrt{14}$

Let a line with direction ratios $a,-4a,-7$ 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 $x-y+z=0$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to ____.

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

1. A.$\frac{\sqrt{3}}{2}$
2. B.$\sqrt{3}$
3. C.$2\sqrt{3}$
4. D.3

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?

1. A.$(6x+5y-12{)}^2+4(3x+7y-8{)}^2=1,z=6-2x-3y$
2. B.$(5x+6y-12{)}^2+4(3x+5y-9{)}^2=1,z=6-2x-3y$
3. C.$(6x+5y-14{)}^2+9(3x+5y-7{)}^2=1,z=6-2x-3y$
4. D.$(5x+6y-14{)}^2+9(3x+7y-8{)}^2=1,z=6-2x-3y$

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 $\mathrm{PRQS}$. If the direction ratios of the diagonal $\mathrm{RS}$ are $\alpha ,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then ${\alpha }^2+{\beta }^2$ is equal to ____.

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

1. A.$(0,-2,-2)$
2. B.$(-5,0,-1)$
3. C.$(3,-1,0)$
4. D.$(0,4,5)$

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 A, B, C making the intercepts $\alpha ,\beta ,\gamma$. If V is the volume of the tetrahedron OABC, where O is the origin, and $p=\alpha +\beta +\gamma$, then the ordered pair $(V,p)$ is equal to :

1. A.$(48,-13)$
2. B.$(24,-13)$
3. C.$(48,11)$
4. D.$(24,-5)$

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 :

1. A.$\frac{1}{11}$
2. B.$\frac{5}{11}$
3. C.6
4. D.7

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 _________.

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 $\mathrm{P}$ in this line, then $a+{\sum }_{i=1}^3{\alpha }_i$ is equal to :

1. A.$7$
2. B.$8$
3. C.$12$
4. D.$14$

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 :

1. A.$\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0$
2. B.$\frac{1}{\sqrt{2}},\pm \frac{1}{\sqrt{2}},0$
3. C.$\frac{1}{\sqrt{5}},-\frac{2}{\sqrt{5}},0$
4. D.$\frac{1}{2},-\frac{\sqrt{3}}{2},0$

If the system of linear equations. $8x+y+4z=-2$ $x+y+z=0$ $\lambda x-3y=\mu$ has infinitely many solutions, then the distance of the point $\left(\lambda ,\mu ,-\frac{1}{2}\right)$ from the plane $8x+y+4z+2=0$ is :

1. A.$3\sqrt{5}$
2. B.4
3. C.$\frac{26}{9}$
4. D.$\frac{10}{3}$

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 :

1. A.$2\sqrt{29}$
2. B.1
3. C.$\sqrt{\frac{37}{29}}$
4. D.$\frac{\sqrt{29}}{2}$

A plane $E$ is perpendicular to the two planes $2x-2y+z=0$ and $x-y+2z=4$, and passes through the point $P(1,-1,1)$. If the distance of the plane $E$ from the point $Q(a,a,2)$ is $3\sqrt{2}$, then $(PQ{)}^2$ is equal to :

1. A.9
2. B.12
3. C.21
4. D.33

The distance of the point $(3,2,-1)$ from the plane $3x-y+4z+1=0$ along the line $\frac{2-x}{2}=\frac{y-3}{2}=\frac{z+1}{1}$ is equal to :

1. A.9
2. B.6
3. C.3
4. D.2

Let the plane $ax+by+cz=d$ pass through $(2,3,-5)$ and is perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$. If $a,b,c,d$ are integers $d>0$ and $\gcd (|a|,|b|,|c|,d)=1$, then the value of $a+7b+c+20d$ is equal to :

1. A.$18$
2. B.$20$
3. C.$24$
4. D.$22$

Let the image of the point $P(1,2,3)$ in the line $L:\frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be Q. Let $R(\alpha ,\beta ,\gamma )$ be a point that divides internally the line segment PQ in the ratio $1:3$. Then the value of $22(\alpha +\beta +\gamma )$ is equal to ________.

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 ____.

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 :

1. A.$\frac{18}{\sqrt{5}}$
2. B.$\frac{22}{3\sqrt{5}}$
3. C.$\frac{46}{3\sqrt{5}}$
4. D.$6\sqrt{3}$

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 $3x+4y+12z+23=0$ is :

1. A.$5$
2. B.$\frac{50}{13}$
3. C.$4$
4. D.$\frac{63}{13}$

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 :

1. A.$\frac{a}{8}=\frac{b}{5}=\frac{c}{-4}$
2. B.$\frac{a}{4}=\frac{b}{5}=\frac{c}{-2}$
3. C.$\frac{a}{8}=-\frac{b}{5}=\frac{c}{4}$
4. D.$\frac{a}{4}=\frac{b}{5}=\frac{c}{2}$

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 :

1. A.$(2,-2,0)$
2. B.$(-2,2,0)$
3. C.$(1,0,2)$
4. D.$(-1,0,-2)$