Basics of 3D Geometry

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

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 $A(20,15), B(0,0) \&\hspace{0.17em}C(-36,15)$.

Find the circum radius of the triangle formed by the points $A\left(0,0,0\right), B\left(0,0,12\right) \text{and} C\left(3,4,0\right)$.

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

$∆ABC$ has vertices at $A\equiv \left(2, 3, 5\right), B\equiv (-1, 3, 2)$ and $C\equiv \left(\lambda , 5, \mu \right).$ If the median through $A$ is equally inclined to the axes, then the values of $\lambda$ and $\mu$ respectively are

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 points $P\left(4, 5, x\right), Q\left(3, y, 4\right)$ and $R\left(5, 8, 0\right)$ are collinear, then the value of $x+y$ 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 $Z_1+Z_2=$

Find the ratio on which the plane $3\text{x}+4\text{y}-5\text{z}=1$ divides the line segment joining the points $\left(-2,4,-6\right)$ and $\left(3,-5,8\right)$.

If $P\left(x, y, z\right)$ is a point on the line segment joining $Q\left(2, 2, 4\right)$ and $R\left(3, 5, 6\right)$ such that the projections of $\overrightarrow{OP}$ on the axes are $\frac{13}{5}, \frac{19}{5}, \frac{26}{5}$ respectively, then $P$ divides $QR$ in the ratio

The value of x so that the point $\left(\text{6, 5,}-3\right)$ is at distance of 13 units from the point $\left(\text{x,}-7\text{, 0} \right)$is

If $\alpha \text{, }\beta \text{ and }\gamma$ are the angles which a directed line makes with the positive directions of the co-ordinates axes, then find the value of  ${\text{sin}}^2\alpha +{\text{sin}}^2\beta +{\text{sin}}^2\gamma$.

A point R with x-co-ordinate 4 lies on the line segment joining the points $P\left(2,-3,4\right)$ and $Q\left(8,0,10\right)$. Find the co-ordinates of the point $R.$