Basics of 3D Geometry

The incentre of the triangle with vertices $\left(1\text{,}\sqrt{3}\right)$, $(0, 0)$ and $(2, 0)$ is

If $x$-coordinate of a point $P$ on the line joining the points $Q\left(2,2,1\right)$ and $R\left(5,2,-2\right)$ is $4$, then the $y$-coordinate of $P=$

If the point $\left(a, 8,-2\right)$ divides the line segment joining the points $\left(1,4,6\right)$ and $\left(5,2,10\right)$ in the ratio $m:n$ then $\frac{2m}{n}-\frac{a}{3}=$

Find the excenter opposite to vertex $C$ of triangle with vertices $A(3,4), B(0,0) \&\hspace{0.17em}C(-3,4)$.

Find the excenter opposite to vertex $B$ of triangle with vertices $A(3,4), B(0,0) \&\hspace{0.17em}C(-3,4)$.

Find the excenter opposite to vertex $C$ of triangle with vertices $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

Find the excenter opposite to vertex $B$ of triangle with vertices $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

Find the excenter opposite to vertex $A$ of triangle with vertices $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

The position vectors of $A$ and $B$ are $(\hat{i}+\hat{j}+\hat{k})$ and $\left(\frac{1}{3}\hat{j}+\frac{1}{3}\hat{k}\right)$. If  $B$ divides the line $A C$ in the ratio $2 : 1,$ then position vector of $C$ is

If the orthocenter and circumcenter of a triangle respectively are $\left(3,-4,2\right)$ and $\left(2,1,3\right),$ then its centroid is______.

The harmonic conjugate of the point $(2, 3, 4)$ with respect to the points $(3, -2, 2)$ and $(6, -17, -4)$ is

The incenter of the triangle formed by the points $(0, 0, 0), (3, 0, 0), (0, 4, 0)$ is

$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+$ $\hat{j}+\hat{k}$ respectively in the ratio $2: 1$ externally. $S$ divides$PQ$ internally in the ratio $2: 1$. Then the position vector of the midpoint of the line joining $R$ and $S$ is

If the origin is the centroid of the triangle for which $-2,3,4$ and $3,-1,5$ are two vertices, then the third vertex is _______

The perimeter of the triangle with verities at (1,0,0).(0,1,0) and (0,0,1) is

The ratio in which the line joining the points $\left(2,4,5\right)$ and $\left(3,5,-4\right)$ is divided by the $yz$ plane is

The points $\left(5,2,4\right),\mathrm{ }\left(6,-1,2\right)$ and $\left(8,-7,k\right)$ are collinear, if $k$ is equal to

The ratio in which $yz$-palne divides the line segment joining $\left(-\mathrm{3,4},-2\right)\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\left(\mathrm{2,1},3\right)\mathrm{i}\mathrm{s}$

The ratio in which the line joining $\left(2,-4,3\right)$ and $\left(-4,5,-6\right)$ is divided by the plane $3x+2y+z-4=0$ is

If $Z_1$ and $Z_2$ are $z-$ coordinates of the points of trisection of the segment joining the points $A(2,1,4), B(-1,3,6)$ then $Z_1+Z_2=$