3d geometry system practice · Mixed difficulty

If the projection of a line segment on $x, y$ and $z$ axes are respectively 3,4 and 5 , then the length of the line segment is

If the origin is the centroid of a $\Delta A B C$ having vertices $A(a, 1,3), B(-2, b,-5)$ and $C(4,7, c)$, then

Given that ${A}(3,2,-4), {B}(5,4-6)$ and ${C}(9,8,-10)$ are collinear. Ratio in which $B$ divides $AC$ is $1:{m}$. The value of ${m}$ is

Vertices of a variable triangle are $(3,4),(5 \cos \theta, 5 \sin \theta)$ and $(5 \sin \theta,-5 \cos \theta)$, where $\theta \in R$. Locus of its orthocentre is

If the projections of a line on the axes are $6,2,3$, then its direction cosines are

The locus of a points which moves such that the sum of the squares of its distance from three vertices of the triangle is constant is a/an

The reflection of the point $(\alpha, \beta, \gamma)$ in the XY-plane is

Assertion: The coordinates of the point which divides the join of ${A}(2,-1,4)$ and ${B}(4,3,2)$ in the ratio $2: 3$ externally is $C(-2,-9,8)$

Reason: If ${P}\left({x}_1, {y}_1, {z}_1\right)$ and ${Q}\left({x}_2, {y}_2, {z}_2\right)$ be two points, and let ${R}$ be a point on ${PQ}$ produced dividing it externally in the ratio ${m}_1: {m}_2$. Then the coordinates of ${R}$ are

$\left(\frac{{m}_1 {x}_2-{m}_2 {x}_1}{{m}_1-{m}_2}, \frac{{m}_1 {y}_2-{m}_2 {y}_1}{{m}_1-{m}_2}, \frac{{m}_1 {z}_2-{m}_2 {z}_1}{{m}_{{1}}-{m}_2}\right)$

The ratio in which the $x$-axis divides the line segment joining $\left( {3,6} \right)$ and $\left( {4, - 3} \right)$ is

The position of a moving point in the xy plane at time t is given by $\left( {u\cos \alpha .t,\;u\sin \alpha .t - \frac{1}{2}g\;{t^2}} \right),$ where $u,\alpha ,g$ are constants. The locus of the moving point is

The perimeter of triangle with vertices $(1, 0, 0), (0, 1, 0)$ and $(0, 0, 1)$ is $k \sqrt{2},$ where the value of $k$ is _______.

The locus of a point $P$ which divides the line joining $(1,0)$ and $(2 \cos \theta, 2 \sin \theta)$ internally in the ratio $2: 3$ for all $\theta$ is

The distance between the points $(1, 4, 5)$ and $(2, 2, 3)$ is___________.

The line joining $A\left( {b\cos \mathrm{\alpha },b\sin \mathrm{\alpha }} \right)$ and $B\left( {a\cos \mathrm{\beta },a\sin \mathrm{\beta }} \right)$ is produced to the point $M\left( {x,y} \right)$ so that $AM:MB = b:a$, then

$x\cos \left( {\frac{{\mathrm{\alpha } + \mathrm{\beta }}}{2}} \right) + y\sin \left( {\frac{{\mathrm{\alpha } + \mathrm{\beta }}}{2}} \right)$ is

If the centroid of the triangle formed by the points (0, 0), $\left( {\cos \mathrm{\theta },\sin \mathrm{\theta }} \right)$ and $\left( {\sin \mathrm{\theta },\; - \cos \mathrm{\theta }} \right)$ lies on the line $y = 2x$, then $\mathrm{\theta }$ is equal to

The distance between the parallel lines given by $(x+7 y)^2+4 \sqrt{2}(x+7 y)-42=0$ is

The equation of the plane containing the points ${A}(1,0,1)$ and ${B}(3,1,2)$ and parallel to the line joining the origin to the point ${C}(1,-1,2)$ is

If the distance between (2, 3) and $\left( { - 5,\;2} \right)$ is equal to the distance between $\left( {x,\;2} \right)$ and (1, 3), then the values of $x$ are

The direction ratios of the line $x-y+z-5=0=x-3 y-6$ are:

$L$ is the foot of the perpendicular drawn from a point $P(3,4,5)$ on the $X Y$-plane. The coordinates of point $L$ are

Three vertices of a parallelogram $\mathrm{ABCD}$ are $\mathrm{A}(3,-1,2), \mathrm{B}(1,2,-4)$ and $\mathrm{C}(-1,1,2)$. The coordinates of fourth vertex $\mathrm{D}$ are

Coordinates of the vertices $B$ and $C$ of the base of a triangle $A B C$ are $(-a, 0)$ and $(a, 0)$ respectively. If $C-B=\pi / 3$, the vertex $A$ lies on the curve

Let $A(a,\,0),\,\,B(b,\,2b + 1)$ and $C(0,\,b),\,\,b \ne 0,\,\,\left| b \right| \ne 1$, be points such that the area of $\Delta ABC$ is 1 sq. unit, then the sum of all possible values of $a$ is

A plane is parallel to XZ-plane, so it is perpendicular to

The ordered pair $(\lambda, \mu)$ such that the points $(\lambda, \mu,-6),(3,2,-4)$ and $(9,8,-10)$ become collinear is

If the points $(-1,2,-3),(4, a, 1)$ and $(b, 8,5)$ are collinear, then $a$ and $b$ are respectively equal to

Locus of the centroid of triangle whose vertices are $\left( {a\cos \;t,\;a\sin t} \right),\left( {b\sin t, - b\cos t} \right)$ and (1,0), where t is a parameter, is

If the centroid of the triangle formed by the points $\left( {a,b} \right),\left( {b,c} \right)$ and $\left( {c,a} \right)$ is at the origin, then ${a^3} + {b^3} + {c^3} =$

The coordinates of three vertices of a quadrilateral in order are (6,1),(7,2) and (-1,0). If the area of the quadrilateral is 4 square units, then the locus of the fourth vertex is

$A(3,2,0), B(5,3,2)$ and $C(-9,6,-3)$ are the vertices of a triangle $A B C$. If the bisector of $\angle A B C$ meets $\mathrm{BC}$ at $\mathrm{D}$, then coordinates of $D$ are