Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 4: Exercise-4

If $V$ be the volume of a tetrahedron and $V^{\text{'}}$ be the volume of the tetrahedron formed by the centroids and $V=k{V}^{\text{'}}$, then find the value of $k$.

Through a point $P(h,k,\ell )$ a plane is drawn at right angles to $OP$ to meet the co-ordinate axes in $A, B$ and $C$. If $OP=p$, show that the area of $∆ABC$ is $\left|\frac{p^5}{2hk\ell }\right|$, where $O$ is the origin.

If $A\left(\overrightarrow{a}\right), B\left(\overrightarrow{b}\right)$ and $C\left(\overrightarrow{c}\right)$ are three non-collinear points and origin does not lie in the plane of the points $A, B$ and $C,$ then for any point $P\left(\overrightarrow{p}\right)$ in the plane of the $\mathit{∆}ABC$, prove that;

$\left(\mathrm{i}\right)$ $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\overrightarrow{p}·\left(\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}\right)$

Prove that the equation of the sphere which passes through the points $\left(1,0,0\right), \left(0,1,0\right)$ and $\left(0,0,1\right)$ and having radius as small as possible is $3\Sigma x^2-2\Sigma x-1=0$

Find the equation of the sphere which has centre at the origin and touches the line $2\left(x+1\right)=2-y=z+3 .$

A mirror and a source of light are situated at the origin $O$ and at a point on $OX$($x-$ axis), respectively. A ray of light from the source strikes the mirror and is reflected. If the $\mathrm{Dr}\text{'}\mathrm{s}$ of the normal to the plane are $1, -1, 1$, then find $\mathrm{d}.\mathrm{c}\text{'}\mathrm{s}$ of the reflected ray.

A variable plane $lx+my+nz=p$ (where $l, m\mathit{,}\mathit{ }n$ are direction cosines) intersects with co-ordinate axes at points $A, B$ and $C$, respectively, show that the foot of normal on the plane from origin is the orthocentre of triangle $ABC$ and hence find the coordinates of circumcentre of triangle $ABC$.

A rectangle whose vertices are $(5,3,-3),(5,9,9),(0,5,11)$ and $(0,-1,-1)$ is rotated about its diagonal (whose direction cosines are $\left(\frac{1}{3},\frac{2}{3},\frac{2}{3}\right)$ in such a way that new position of rectangle is perpendicular to its old position find the coordinates of new position of the vertices whose position is changed.