182 solved questions on Three Dimensional Geometry, ranging from easy to JEE-Advanced-flavour hard. Click any to see the full solution.
Let the line $l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}$ meet the plane $P: x+2 y+3 z=4$ at the point $(\alpha, \beta, \gamma)$. If the angle between the…
View solution →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…
View solution →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, -…
View solution →The acute angle between the planes P1 and P2, when P1 and P2 are the planes passing through the intersection of the planes $5x + 8y + 13z - 29 = 0$ and $8x - 7y + z - 20 = 0$ and…
View solution →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 :
View solution →The shortest distance between the lines ${{x - 1} \over 0} = {{y + 1} \over { - 1}} = {z \over 1}$ and x + y + z + 1 = 0, 2x – y + z + 3 = 0 is :
View solution →The shortest distance, between lines $L_1$ and $L_2$, where $L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$ is the line, passing through the points…
View solution →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 ${{\sqrt 7 } \over {2\sqrt 2…
View solution →If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum…
View solution →The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point :
View solution →If the shortest distance between the lines $$\begin{array}{ll} L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda \in \mathbb{R} \\ L_2:…
View solution →Let the lines $${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$$ $${L_2}:\overrightarrow r = \left( {\widehat i +…
View solution →If $A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)$ and $D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral $A B…
View solution →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?
View solution →If the equation of the plane passing through the line of intersection of the planes $2 x-y+z=3,4 x-3 y+5 z+9=0$ and parallel to the line…
View solution →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…
View solution →The plane passing through the line $L: l x-y+3(1-l) z=1, x+2 y-z=2$ and perpendicular to the plane $3 x+2 y+z=6$ is $3 x-8 y+7 z=4$. If $\theta$ is the acute angle between the…
View solution →The square of the distance of the point of intersection of the line ${{x - 1} \over 2} = {{y - 2} \over 3} = {{z + 1} \over 6}$ and the plane $2x - y + z = 6$ from the point…
View solution →Let the image of the point $\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$ in the plane $x-2 y+z-2=0$ be P. If the distance of the point $Q(6,-2, \alpha), \alpha 0$, from…
View solution →The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :
View solution →If the shortest between the lines ${{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}$ and ${{x - \lambda } \over 3} = {{y - 2\sqrt 6 } \over 4} = {{z…
View solution →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…
View solution →For real numbers $\alpha$ and $\beta$ $\ne$ 0, if the point of intersection of the straight lines${{x - \alpha } \over 1} = {{y - 1} \over 2} = {{z - 1} \over 3}$ and ${{x - 4}…
View solution →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,…
View solution →Let the equation of the plane passing through the line $x - 2y - z - 5 = 0 = x + y + 3z - 5$ and parallel to the line $x + y + 2z - 7 = 0 = 2x + 3y + z - 2$ be $ax + by + cz =…
View solution →If the foot of the perpendicular from point (4, 3, 8) on the line ${L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}$, l $\ne$ 0 is (3, 5, 7), then the shortest…
View solution →The equation of the plane passing through the line of intersection of the planes $\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$ and…
View solution →The shortest distance between the lines ${{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}$ and ${{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}$, is :
View solution →The foot of the perpendicular from a point on the circle $x^{2}+y^{2}=1, z=0$ to the plane $2 x+3 y+z=6$ lies on which one of the following curves?
View solution →Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3 \hat{i}+2 \hat{j}+2 \hat{k}$. If the distance of the point Q from the point…
View solution →If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all…
View solution →The shortest distance between the lines ${{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2}$ and ${{x - 4} \over 1} = {{y - 1} \over 2} = {{z + 3} \over 0}$ is :
View solution →Let the equation of the plane P containing the line $x+10=\frac{8-y}{2}=z$ be $ax+by+3z=2(a+b)$ and the distance of the plane $P$ from the point (1, 27, 7) be $c$. Then…
View solution →If the equation of a plane P, passing through the intersection of the planes, x + 4y - z + 7 = 0 and 3x + y + 5z = 8 is ax + by + 6z = 15 for some a, b $\in$ R, then the distance…
View solution →Let a plane P pass through the point (3, 7, $-$7) and contain the line, ${{x - 2} \over { - 3}} = {{y - 3} \over 2} = {{z + 2} \over 1}$. If distance of the plane P from the…
View solution →Let $\mathrm{P}$ 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…
View solution →Consider the lines $L_1$ and $L_2$ given by ${L_1}:{{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 2} \over 2}$ ${L_2}:{{x - 2} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$. A…
View solution →Let a plane P contain two lines $$\overrightarrow r = \widehat i + \lambda \left( {\widehat i + \widehat j} \right)$$, $\lambda \in R$ and $$\overrightarrow r = - \widehat j + \mu…
View solution →If a point $\mathrm{P}(\alpha, \beta, \gamma)$ satisfying $$\left( {\matrix{ \alpha & \beta & \gamma \cr } } \right)\left( {\matrix{ 2 & {10} & 8 \cr 9 & 3 & 8 \cr 8 & 4 & 8 \cr }…
View solution →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 $\mathrm{P}$ be the plane containing…
View solution →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…
View solution →If the line of intersection of the planes $a x+b y=3$ and $a x+b y+c z=0$, a $0$ makes an angle $30^{\circ}$ with the plane $y-z+2=0$, then the direction cosines of the line are :
View solution →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…
View solution →Let a line with direction ratios $a,-4 a,-7$ be perpendicular to the lines with direction ratios $3,-1,2 b$ and $b, a,-2$. If the point of intersection of the line…
View solution →The shortest distance between the lines $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is:
View solution →Let $\mathrm{L}_1: \frac{x-1}{1}=\frac{y-2}{-1}=\frac{z-1}{2}$ and $\mathrm{L}_2: \frac{x+1}{-1}=\frac{y-2}{2}=\frac{z}{1}$ be two lines. Let $L_3$ be a line passing through the…
View solution →The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is
View solution →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 $\mathrm{L}$ be the…
View solution →Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3…
View solution →The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $-$ n = 0 and mn + nl + lm = 0, is :
View solution →Let the plane $P:\overrightarrow r \,.\,\overrightarrow a = d$ contain the line of intersection of two planes $$\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat…
View solution →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…
View solution →If the two lines ${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$ and ${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}$ are perpendicular, then an…
View solution →Let the lines $l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$ and $l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13$ be coplanar. If the point $\mathrm{P}(a, b, c)$ on $l_{1}$ is…
View solution →The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3, $-$4, $-$5) and R(2, $-$3, 1) and the plane 2x + y + z = 7, is equal to…
View solution →Let P be a plane passing through the points (1, 0, 1), (1, $-$2, 1) and (0, 1, $-$2). Let a vector $\overrightarrow a = \alpha \widehat i + \beta \widehat j + \gamma \widehat k$…
View solution →Suppose, the line ${{x - 2} \over \alpha } = {{y - 2} \over { - 5}} = {{z + 2} \over 2}$ lies on the plane $x + 3y - 2z + \beta = 0$. Then $(\alpha + \beta )$ is equal to _______.
View solution →Consider the three planesP1 : 3x + 15y + 21z = 9,P2 : x $-$ 3y $-$ z = 5, and P3 : 2x + 10y + 14z = 5Then, which one of the following is true?
View solution →The shortest distance between the lines ${{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}$ and ${{x - 1} \over 2} = {{y - 2} \over 1} = {{z - 6} \over { - 3}}$ is :
View solution →If the shortest distance between the lines ${{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }$ and ${{x - 2} \over 1} = {{y - 4} \over 4} = {{z - 5} \over 5}$ is…
View solution →If the lines ${{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over 1}$ and ${{x - a} \over 2} = {{y + 2} \over 3} = {{z - 3} \over 1}$ intersect at the point P, then the…
View solution →If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :
View solution →A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a, b, c)$, where $a, b, c$ are non-negative integers, on the line $A B$ lies at a distance of 21 units, from the…
View solution →The distance of the point $(7,10,11)$ from the line $\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6}$ is
View solution →The mirror image of the point (1, 2, 3) in a plane is $\left( { - {7 \over 3}, - {4 \over 3}, - {1 \over 3}} \right)$. Which of the following points lies on this plane ?
View solution →The distance of line $3y - 2z - 1 = 0 = 3x - z + 4$ from the point (2, $-$1, 6) is :
View solution →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…
View solution →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 ${\pi \over 2}$, then the plane after the rotation…
View solution →Let the point, on the line passing through the points $P(1,-2,3)$ and $Q(5,-4,7)$, farther from the origin and at a distance of 9 units from the point $P$, be $(\alpha, \beta,…
View solution →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 :
View solution →Let $P Q R$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $\mathrm{PQ}$. The distance of the centroid of $\triangle \mathrm{PQR}$ from the point of…
View solution →If equation of the plane that contains the point $(-2,3,5)$ and is perpendicular to each of the planes $2 x+4 y+5 z=8$ and $3 x-2 y+3 z=5$ is $\alpha x+\beta y+\gamma z+97=0$ then…
View solution →Let the plane passing through the point ($-$1, 0, $-$2) and perpendicular to each of the planes 2x + y $-$ z = 2 and x $-$ y $-$ z = 3 be ax + by + cz + 8 = 0. Then the value of a…
View solution →The lines x = ay $-$ 1 = z $-$ 2 and x = 3y $-$ 2 = bz $-$ 2, (ab $\ne$ 0) are coplanar, if :
View solution →If the equation of the plane containing the line $x+2 y+3 z-4=0=2 x+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…
View solution →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,…
View solution →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 :
View solution →Let a line having direction ratios, 1, $-$4, 2 intersect the lines ${{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z + 2} \over 1}$ and ${x \over 2} = {{y - 7} \over 3} = {z \over…
View solution →A line 'l' passing through origin is perpendicular to the lines$${l_1}:\overrightarrow r = (3 + t)\widehat i + ( - 1 + 2t)\widehat j + (4 + 2t)\widehat k$$$${l_2}:\overrightarrow…
View solution →Let the foot of the perpendicular from the point (1, 2, 4) on the line ${{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1} \over 3}$ be P. Then the distance of P from the plane $3x…
View solution →Let $$L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}$$, $$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \hat{i}+\hat{j}+p \hat{k}), \mu…
View solution →Let the image of the point P(1, 2, 3) in the line $L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}$ be Q. Let R ($\alpha$, $\beta$, $\gamma$) be a point that divides…
View solution →If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of…
View solution →If the line $\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$, then $4 \lambda+9 \mu$ is equal to :
View solution →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 :
View solution →If for some $\alpha$ $\in$ R, the lines L1 : ${{x + 1} \over 2} = {{y - 2} \over { - 1}} = {{z - 1} \over 1}$ and L2 : ${{x + 2} \over \alpha } = {{y + 1} \over {5 - \alpha }} =…
View solution →Let the line $\mathrm{L}$ intersect the lines $x-2=-y=z-1,2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following…
View solution →Let the plane P: $4 x-y+z=10$ be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane $x+y-z=4$. If $\alpha$ is the distance of the point $(2,3,-4)$…
View solution →If $(2,3,9),(5,2,1),(1, \lambda, 8)$ and $(\lambda, 2,3)$ are coplanar, then the product of all possible values of $\lambda$ is:
View solution →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 $4 a x-y+5 z-7 a=0=2 x-5 y-z-3,…
View solution →A vector $\vec{a}$ is parallel to the line of intersection of the plane determined by the vectors $\hat{i}, \hat{i}+\hat{j}$ and the plane determined by the vectors…
View solution →If $\lambda_{1}
View solution →The shortest distance between the lines $x+1=2y=-12z$ and $x=y+2=6z-6$ is :
View solution →Let P be a plane containing the line ${{x - 1} \over 3} = {{y + 6} \over 4} = {{z + 5} \over 2}$ and parallel to the line ${{x - 1} \over 4} = {{y - 2} \over { - 3}} = {{z + 5}…
View solution →Let p be the plane passing through the intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5$$ and $$\overrightarrow r…
View solution →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…
View solution →If $\mathrm{d}_1$ is the shortest distance between the lines $x+1=2 y=-12 z, x=y+2=6 z-6$ and $\mathrm{d}_2$ is the shortest distance between the lines…
View solution →Let a, b$\in$R. If the mirror image of the point P(a, 6, 9) with respect to the line ${{x - 3} \over 7} = {{y - 2} \over 5} = {{z - 1} \over { - 9}}$ is (20, b, $-$a$-$9), then |…
View solution →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…
View solution →The shortest distance between the lines ${{x - 3} \over 3} = {{y - 8} \over { - 1}} = {{z - 3} \over 1}$ and ${{x + 3} \over { - 3}} = {{y + 7} \over 2} = {{z - 6} \over 4}$ is :
View solution →For $\mathrm{a}, \mathrm{b} \in \mathbb{Z}$ and $|\mathrm{a}-\mathrm{b}| \leq 10$, let the angle between the plane $\mathrm{P}: \mathrm{ax}+y-\mathrm{z}=\mathrm{b}$ and the line…
View solution →The shortest distance between the lines ${{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}$ and ${{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}$ is :
View solution →Let d be the distance between the foot of perpendiculars of the points P(1, 2, $-$1) and Q(2, $-$1, 3) on the plane $-$x + y + z = 1. Then d2 is equal to ___________.
View solution →Let ${{x - 2} \over 3} = {{y + 1} \over { - 2}} = {{z + 3} \over { - 1}}$ lie on the plane $px - qy + z = 5$, for some p, q $\in$ R. The shortest distance of the plane from the…
View solution →A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a plane passing through the…
View solution →Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes $$\overrightarrow r \,.\,\left( {\widehat i + \widehat j + 4\widehat k} \right) =…
View solution →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 :
View solution →Let $\mathrm{P}_{1}$ be the plane $3 x-y-7 z=11$ and $\mathrm{P}_{2}$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$. If the foot of the perpendicular…
View solution →Let l1 be the line in xy-plane with x and y intercepts ${1 \over 8}$ and ${1 \over {4\sqrt 2 }}$ respectively, and l2 be the line in zx-plane with x and z intercepts $- {1 \over…
View solution →Let the acute angle bisector of the two planes x $-$ 2y $-$ 2z + 1 = 0 and 2x $-$ 3y $-$ 6z + 1 = 0 be the plane P. Then which of the following points lies on P?
View solution →Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is :
View solution →Let the system of linear equations $-x+2 y-9 z=7$ $-x+3 y+7 z=9$ $-2 x+y+5 z=8$ $-3 x+y+13 z=\lambda$ has a unique solution $x=\alpha, y=\beta, z=\gamma$. Then the distance of the…
View solution →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 $2x+y-z+1=0$. If d is the distance of P from the point…
View solution →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 $\mathrm{P}$ passes…
View solution →If the equation of the plane passing through the line of intersection of the planes 2x $-$ 7y + 4z $-$ 3 = 0, 3x $-$ 5y + 4z + 11 = 0 and the point ($-$2, 1, 3) is ax + by + cz…
View solution →Equation of a plane at a distance $\sqrt {{2 \over {21}}}$ from the origin, which contains the line of intersection of the planes x $-$ y $-$ z $-$ 1 = 0 and 2x + y $-$ 3z + 4 =…
View solution →Let a line $l$ pass through the origin and be perpendicular to the lines $$l_{1}: \vec{r}=(\hat{\imath}-11 \hat{\jmath}-7 \hat{k})+\lambda(\hat{i}+2 \hat{\jmath}+3 \hat{k}),…
View solution →Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations l + m $-$ n = 0 and l2 + m2 $-$ n2 = 0. Then the value of sin4$\alpha$ + cos4$\alpha$ is :
View solution →Consider the line L given by the equation ${{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}$. Let Q be the mirror image of the point (2, 3, $-$1) with respect to L. Let a…
View solution →Let P be the point of intersection of the line ${{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2}$ and the plane $x+y+z=2$. If the distance of the point P from the plane…
View solution →If the lines $$\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)$$ and $$\overrightarrow r = \left(…
View solution →If the shortest distance between the lines $\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the…
View solution →If the lines ${{x - k} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}$ and ${{x + 1} \over 3} = {{y + 2} \over 2} = {{z + 3} \over 1}$ are co-planar, then the value of k is…
View solution →The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is ____________.
View solution →A plane P meets the coordinate axes at A, B and C respectively. The centroid of $\Delta$ABC is given to be (1, 1, 2). Then the equation of the line through this centroid and…
View solution →Let the line $L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}$ intersect the plane $2 x+y+3 z=16$ at the point $P$. Let the point $Q$ be the foot of perpendicular from the point…
View solution →If the distance between the plane, 23x – 10y – 2z + 48 = 0 and the plane containing the lines ${{x + 1} \over 2} = {{y - 3} \over 4} = {{z + 1} \over 3}$ and $${{x + 3} \over…
View solution →Let Q be the foot of the perpendicular from the point P(7, $-$2, 13) on the plane containing the lines ${{x + 1} \over 6} = {{y - 1} \over 7} = {{z - 3} \over 8}$ and ${{x - 1}…
View solution →The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $\mathrm{P}(2,-10,1)$, is :
View solution →The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the point :
View solution →Let $\overrightarrow a = \widehat i + \widehat j + 2\widehat k$, $\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k$ and $\overrightarrow c = \widehat i - \widehat j +…
View solution →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 :
View solution →Let the shortest distance between the lines $L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0$ and $L_{1}: x+1=y-1=4-z$ be $2 \sqrt{6}$. If $(\alpha,…
View solution →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 :
View solution →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…
View solution →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 $\cos…
View solution →Let P be an arbitrary point having sum of the squares of the distances from the planes x + y + z = 0, lx $-$ nz = 0 and x $-$ 2y + z = 0, equal to 9. If the locus of the point P…
View solution →Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, $-$1, $-$1), parallel to the line PQ meets the plane…
View solution →The equation of the planes parallel to the plane x $-$ 2y + 2z $-$ 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b $-$ d) = k(c $-$ a),…
View solution →Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $Q(\alpha, \beta, \gamma)$. Then the distance of…
View solution →If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line ${{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}$ is $\alpha$,…
View solution →A plane $E$ is perpendicular to the two planes $2 x-2 y+z=0$ and $x-y+2 z=4$, and passes through the point $P(1,-1,1)$. If the distance of the plane $E$ from the point $Q(a, a,…
View solution →Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2 \sqrt{17}$ from the foot of perpendicular…
View solution →The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, is :
View solution →If the foot of the perpendicular from the point $\mathrm{A}(-1,4,3)$ on the plane $\mathrm{P}: 2 x+\mathrm{m} y+\mathrm{n} z=4$, is $\left(-2, \frac{7}{2}, \frac{3}{2}\right)$,…
View solution →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 $2 x+3…
View solution →Let $\mathrm{d}$ be the distance of the point of intersection of the lines $\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ from the point…
View solution →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 :
View solution →If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x-3 y+ 2 z-1=0=4 x-y+z$ is $\mathrm{A} x+\mathrm{B} y+\mathrm{C} z=1$, then…
View solution →Let the equation of plane passing through the line of intersection of the planes $x+2 y+a z=2$ and $x-y+z=3$ be $5 x-11 y+b z=6 a-1$. For $c \in \mathbb{Z}$, if the distance of…
View solution →Let $${P_1}:\overrightarrow r \,.\,\left( {2\widehat i + \widehat j - 3\widehat k} \right) = 4$$ be a plane. Let P2 be another plane which passes through the points (2, $-$3, 2),…
View solution →Let O be the origin, and M and $\mathrm{N}$ be the points on the lines $\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$ and $\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$ respectively…
View solution →Let $P(x, y, z)$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P=\gamma$; the angle between $O Q$ and the positive $x$-axis be…
View solution →The vector equation of the plane passing through the intersection of the planes $\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1$ and $\overrightarrow…
View solution →A plane passing through the point (3, 1, 1) contains two lines whose direction ratios are 1, –2, 2 and 2, 3, –1 respectively. If this plane also passes through the point…
View solution →The distance of the point $(-1,2,3)$ from the plane $\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$ parallel to the line of the shortest distance between the lines…
View solution →Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $A B C D$. If the diagonal $\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k}$, then the area of the…
View solution →If the plane $P$ passes through the intersection of two mutually perpendicular planes $2 x+k y-5 z=1$ and $3 k x-k y+z=5, k
View solution →The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line ${x \over 2} = {y \over 3} = {z \over { - 6}}$ is :
View solution →The distance of the point (1, 1, 9) from the point of intersection of the line ${{x - 3} \over 1} = {{y - 4} \over 2} = {{z - 5} \over 2}$ and the plane x + y + z = 17 is :
View solution →If the mirror image of the point (1, 3, 5) with respect to the plane 4x $-$ 5y + 2z = 8 is ($\alpha$, $\beta$, $\gamma$), then 5($\alpha$ + $\beta$ + $\gamma$) equals :
View solution →If the shortest distance between the lines $$\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)$$ and…
View solution →Let $\mathrm{Q}$ and $\mathrm{R}$ be the feet of perpendiculars from the point $\mathrm{P}(a, a, a)$ on the lines $x=y, z=1$ and $x=-y, z=-1$ respectively. If $\angle…
View solution →If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression $$3 + {{x - 11} \over…
View solution →The line $l_1$ passes through the point (2, 6, 2) and is perpendicular to the plane $2x+y-2z=10$. Then the shortest distance between the line $l_1$ and the line…
View solution →Let the line of the shortest distance between the lines $$ \begin{aligned} & \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(\hat{i}+2 \hat{j}+3…
View solution →Let $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $\mathrm{L}_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}, \alpha \in \mathbf{R}$, be two lines, which…
View solution →The foot of perpendicular from the origin $\mathrm{O}$ to a plane $\mathrm{P}$ which meets the co-ordinate axes at the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ is $(2,…
View solution →Let the plane $\mathrm{P}: 8 x+\alpha_{1} y+\alpha_{2} z+12=0$ be parallel to the line $\mathrm{L}: \frac{x+2}{2}=\frac{y-3}{3}=\frac{z+4}{5}$. If the intercept of $\mathrm{P}$ on…
View solution →Let $\mathrm{S}$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}$ and…
View solution →Let $\mathrm{P}$ and $\mathrm{Q}$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the point $\mathrm{R}(1,2,3)$. If…
View solution →The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ is :
View solution →If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m$,…
View solution →Let $\mathrm{A}(x, y, z)$ be a point in $x y$-plane, which is equidistant from three points $(0,3,2),(2,0,3)$ and $(0,0,1)$. Let $\mathrm{B}=(1,4,-1)$ and $\mathrm{C}=(2,0,-2)$.…
View solution →Let a straight line $L$ pass through the point $P(2, -1, 3)$ and be perpendicular to the lines $ \frac{x - 1}{2} = \frac{y + 1}{1} = \frac{z - 3}{-2} $ and $ \frac{x - 3}{1} =…
View solution →The line $\mathrm{L}_1$ is parallel to the vector $\overrightarrow{\mathrm{a}}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and passes through the point $(7,6,2)$ and the line $\mathrm{L}_2$…
View solution →Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$-axes, respectively, is half of the angle that this line makes with the positive…
View solution →Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$ be $3 \sqrt{30}$. Then the…
View solution →Let the values of p , for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\overrightarrow{\mathrm{r}}=(\mathrm{p} \hat{i}+2…
View solution →If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x}{1}=\frac{y}{\alpha}=\frac{z-5}{1}$ is $\frac{5}{\sqrt{6}}$, then the sum of…
View solution →If the equation of the line passing through the point $ \left( 0, -\frac{1}{2}, 0 \right) $ and perpendicular to the lines $ \vec{r} = \lambda \left( \hat{i} + a\hat{j} + b\hat{k}…
View solution →Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-\lambda}{3} = \frac{y-4}{4} =…
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