3d geometry system practice · Easy

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 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 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

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:

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

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 ${a^3} + {b^3} + {c^3} =$

$A(3,2,0), B(5,3,2)$ 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

Locus of centroid of the 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

The intercepts of the plane $2 x-3 y+4 z=12$ on the coordinate axes are given by

The ratio in which the line $x + y = 4$ divides the line joining the points $\left( {1,\; - 1} \right)$ and (5, 7) is

The number of lines that can be drawn through the point $(4,-5)$ at a distance 12 from the point $(-2,3)$ is

If the vertices $A$ and $B$ of a triangle $A B C$ are given by $(2,5)$ and $(4,-11)$ and $C$ moves along the line $L_1: 9 x+7 y+4=0$, the locus of the centroid of the triangle $A B C$ is a straight line parallel to

The equation of the sphere whose centre has the position vector $(3 \hat{i}+6 \hat{j}-4 \hat{k})$ and which touches the plane ${r} \cdot(2 \hat{i}-2 \hat{j}-\hat{k})=10$ is

The coordinates of the point of intersection of the line $\frac{x+1}{1}=\frac{y+3}{3}=\frac{z-2}{-2}$ with the plane $3 x+4 y+5 z=25$ are

If a straight line makes an angle of $\frac{\pi}{4}$ with each of $x$ axis and $y$-axis, then the angle made by the line with $z$-axis is

The projection of the line segment joining the points $(-1,0,3)$ and $(2,5,1)$ on the line whose direction ratios are $6,2,3$, is

Points $(-2,4,7),(3,-6,-8)$ and $(1,-2,-2)$ are