Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 3: Level 3

Let $P\left(x,y,z\right)$ be the foot of the perpendicular from the point $Q\left(0,2,3\right)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. If $R$ denotes the point $\left(-3,1,-4\right)$ and $A$ denotes the area of $∆PQR$, then $2A^2$ is equal to

The length intercepted by a line with direction ratios $2,7,-5$ between the lines $\frac{x-5}{3}=\frac{y-7}{-1}=\frac{z+2}{1}$and $\frac{x+3}{-3}=\frac{y-3}{2}=\frac{z-6}{4}$ is 

The direction ratios of a line are $2+\lambda$, $2-\lambda$, $1$ and it makes an angle of $60°$ with $x-$axis. If $\lambda =\frac{-a\pm b\sqrt{2}}{4}$, then $\frac{5a}{b}=$

A plane passing through $(1,1,1)$ cuts positive direction of coordinate axes at $A, B$ and $C$, then the volume of tetrahedron $OABC$satisfies

If the equation of plane passing through the mirror image of a point $\left(2, 3, 1\right)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z+1}{1}$ is $\alpha x+\beta y+\gamma z=24$ then $\alpha +\beta +\gamma$ is equal to:

Let $L$ be the line of intersection of planes $\overrightarrow{r}·(\hat{i}-\hat{j}+2\hat{k})=2$ and $\overrightarrow{r}·(2\hat{i}+\hat{j}-\hat{k})=2.$ If $P(\alpha , \beta , \gamma )$ is the foot of perpendicular on $L$ from the point $(1, 2, 0),$ then the value of $35(\alpha +\beta +\gamma )$ is equal to:

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}.$ Let $Q$ be the mirror image of the point $(2,3,-1)$ with respect to $L.$ Let a plane $P$ be such that it passes through $Q,$ and the line $L$ is perpendicular to $P.$ Then which of the following points is on the plane $P?$

The reflection of the point $\overrightarrow{a}$ in the plane $\overrightarrow{r}·\overrightarrow{n}=q$ is