Three Dimensional Geometry

The equation of the plane passing through the point $(1,2,1)$ and perpendicular to the planes $x + 2y + 2z - 7 = 0$ and $3x + 3y + 2z - 5 = 0$ is

If the lines $x = -1 + s$, $y = 3 - \lambda s$, $z = 1 + \lambda s$ and $x = \dfrac{t}{2}$, $y = 1 + t$, $z = 2 - t$ with parameters s and t, are coplanar, then $\lambda =$

If the lines $\vec{r} = (\hat{i} + m\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (4\hat{i} + \hat{j}) + \mu(5\hat{i} + m\hat{j} + \hat{k})$ intersect each other, then m =

The co-ordinates of the point where the line joining the points $(3,5,-7)$ and $(-2, 1, 8)$ is intersected by the YOZ plane are

The plane $\dfrac{x}{2} + \dfrac{y}{3} + \dfrac{z}{4} = 1$ cuts the co-ordinate axes at the points A, B, C respectively. Then the area of triangle ABC is

Let a plane P pass through the point $(3, 7, -7)$ and contain the line $\dfrac{x - 2}{-3} = \dfrac{y - 3}{2} = \dfrac{z + 2}{1}$. If the distance of the plane P from the origin is d, then $d^2$ is

The co-ordinates of the foot of the perpendicular from the origin to the plane $2x - 3y - 6z = 49$ are...

The shortest distance between the lines, where the first line passes through $(0,0,0)$ and $(2,0,3)$ and the second line passes through $(2,5,0)$ and $(0,4,0)$ is

The point having position vector $4\hat{i} - 11\hat{j} + 2\hat{k}$ lies on the line

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then the sum of values of $\sin^2\alpha + \sin^2\beta + \sin^2\gamma$ and $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$ is ...

The equation of the plane passing through the intersection of planes $2x - y + z = 3$, $4x - 3y + 5z = -9$ and parallel to the line $\dfrac{x+1}{2} = \dfrac{y+3}{4} = \dfrac{z-3}{5}$ is...

The equation of a line passing through a point $(4, -2, 3)$ and perpendicular to the XZ-plane is....

The vector equation of the plane which is at a distance of $5$ units from the origin and normal to the vector $2\hat{i} + \hat{j} - 2\hat{k}$ is

A plane meets the coordinate axes at points A, B and C, such that the centroid of a triangle ABC is $\left(2, -\dfrac{2}{3}, \dfrac{1}{2}\right)$. The perpendicular distance from the origin to this plane is...

If the lines $L_1: \dfrac{x-1}{-3} = \dfrac{y-2}{2k} = \dfrac{z-3}{2}, L_2: \dfrac{x-1}{3k} = \dfrac{y-5}{1} = \dfrac{z-6}{-5}$ are perpendicular to each other, then the equation of a plane containing the line $L_1$ and parallel to the line...

If the lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-2}{1} = \dfrac{y+m}{2} = \dfrac{z-2}{1}$ intersect each other, then the value of $m$ is....

If the points $(1, 1, \mu)$ and $(-3, 0, 1)$ are equidistant from the plane $\vec{r} \cdot (3\hat{i} + 4\hat{j} - 12\hat{k}) + 13 = 0$, then the value of $\mu$ are

If the lines $\dfrac{x-1}{2} = \dfrac{y+1}{3} = \dfrac{z-1}{4}$ and $\dfrac{x-3}{1} = \dfrac{y-c}{2} = \dfrac{z}{1}$ intersect, then the radius of circle $x^2 + y^2 - 4x + 10y + c = 0$ is

If the centroid of a tetrahedron $OABC$ is $(1, 2, -1)$, where O is the origin, $A(a, 2, 3), B(1, b, 2), C(2, 1, c)$ are the other vertices, then the distance of the point $P(a, b, c)$ from the origin is...

The point of intersection of the two lines $\dfrac{x-3}{3} = \dfrac{y-3}{-1}, z - 1 = 0$ and $\dfrac{x-6}{2} = \dfrac{z-1}{3}, y - 2 = 0$ is....

The vector equation of the plane passing through the point $A(-2, 7, 5)$ and parallel to the vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ is ...

A line L is passing through points $\mathrm{A}(1,3,2)$ and $\mathrm{B}(2,2,1)$. If mirror image of point $\mathrm{P}(1,1,-1)$ in the line L is $(x, y, z)$ then $x+y+\mathrm{z}=$

The equation of plane passing through $(1,0,0)$ and $(0,1,0)$ and making an angle $45^{\circ}$ with the plane $x+y-3=0$ is

The co-ordinates of the point in which line joining $(1,1,1)$ and $(2,2,2)$ intersects the plane $x+y+\mathrm{z}=9$ are

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

The distance of the point $(5,3,-1)$ from the plane passing through points $(2,1,0),(3,-2,4)$ and $(1,-3,3)$ is

The mirror image of the point $\mathrm{P}(-1,2,-4)$ in the plane $x-y-2 z+1=0$ is

A triangle ABC is formed by $\mathrm{A}(1,-1,0)$, $B(3,5,3), C(-11,-5,6)$. The equation of internal angle bisector of angle $A$ is

If the foot of the perpendicular drawn from the origin to a plane is $\mathrm{P}(-1,-1,2)$, then equation of the plane is

If the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{6}=1$ cuts the co-ordinate axes at points $A, B, C$ respectively, then area of the triangle ABC is

The angle between lines whose direction cosines satisfy the equation $l+m+n=0$ and $l^2-\mathrm{m}^2-\mathrm{n}^2=0$, is

The line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z}{1}$ intersects the XY plane and the YZ plane at points A and B respectively. The equation of line through the points A and B is

The distance of the point $\mathrm{A}(3,-4,5)$ from the plane $2 x+5 y-6 z=16$ measured along the line $\frac{x}{2}=\frac{y}{1}=\frac{z}{-2}$ is

The direction cosines of a normal to the plane passing through $(4,2,3),(-1,4,2)$ and $(3,2,1)$ are …..

The equation of the plane passing through the point $(1,1,1)$ and through the line of intersection of $x+2 y-z+1=0$ and $3 x-y-4 z+3=0$ is

The lines $\frac{6 x-6}{18}=\frac{y+1}{3}=\frac{z-1}{5} \quad$ and $\frac{3 x+6}{12}=\frac{y-1}{3}=\frac{z+1}{2}$ are $\ldots$

In 3-dimensional space, the equation $x^2-8 x+12=0$ represents ....

The lines $\frac{x-0}{1}=\frac{y-2}{2}=\frac{z+3}{\lambda}$ and $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{\lambda}$ are coplanar and $p$ is the plane containing these lines, then which of following point does not lie on the plane.

If the lines $\frac{1-x}{2}=\frac{7 y+4}{2 \lambda}=\frac{2 z-5}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-1}{7}=\frac{6-\mathrm{z}}{5}$ are at right angle, then the value of $\lambda$ is

Let M and N be foots of the perpendiculars drawn from the point $\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})$ on the lines $x-y=0, \mathrm{z}=1$ and $x+y=0, \mathrm{z}=-1$ respectively and if $\angle \mathrm{MPN}=90^{\circ}$ then $\mathr...

The length of the foot of the perpendicular from the point $\left(1, \frac{3}{2}, 2\right)$ to the plane $2 x-2 y+4 z+17=0$ is

If the line $\frac{x-3}{2}=\frac{y+5}{-1}=\frac{z+2}{2}$ lies in the plane $\alpha x+3 y-z+\beta=0$, then values of $\alpha$ and $\beta$ respectively are ….

The Cartesian equation of the plane $\overline{\mathrm{r}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ is

If the plane $\frac{x}{2}-\frac{y}{3}-\frac{\mathrm{z}}{5}=1$ cuts the co-ordinate axes in points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ respectively, then the area of the triangle $A B C$ is

If the lines $x=a y-1=z-2$ and $x=3 y-2=\mathrm{bz}-2(\mathrm{ab} \neq 0)$ are coplanar, then

The lines $\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(3 \hat{i}-\hat{j})$ and $\overline{\mathrm{r}}=(4 \hat{\mathrm{i}}-\hat{\mathrm{k}})+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}})$ are

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$ intersect, then the value of k is

The equation of the plane containing the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and perpendicular to the plane containing the lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{1}$ and $\frac{x}{3}=\frac{y}{2}=\frac{z}{1}$ is

The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points $\mathrm{Q}(3,-4,-5), \mathrm{R}(2,-3,1)$ and the plane $2 x+y+z=7$ is

The equation of the plane containing the line $\frac{x+1}{2}=\frac{y+2}{1}=\frac{z-2}{3}$ and the point $(1,-1,3)$ is