Embibe Experts Solutions for Chapter: Three Dimensional Geometry, Exercise 1: JEE Advanced Paper 1 - 2013

Let $\alpha , \beta , \gamma , \delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ and $\alpha +\gamma =1$. Suppose the point $\left(3,2,-1\right)$ is the mirror image of the point $\left(1,0,-1\right)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?

Three lines
$L_1:\overrightarrow{r}=\lambda \widehat{i}, \lambda \in R,$
$L_2:\overrightarrow{r}=\widehat{k}+\mu \widehat{j}, \mu \in R$ and
$L_3: \overrightarrow{r}=\hat{i}+\hat{j}+\nu \hat{k}, \nu \in R$
are given. For which point(s) $Q$ and $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P, Q$ and $R$ are collinear?

Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the $z$-axis. Let the distance of $P$ from the $x$-axis be $5$. If $R$ is the image of $P$ in the $xy$-plane, then the length of $PR$ is ______.

The equation of the plane passing through the point $(\mathrm{1,1},1)$ and perpendicular to the planes $2x+y-2z=5$ and $3x-6y-2z=7$ is

Let $L_1$ and $L_2$ be the following straight lines.
$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$
Suppose the straight line $L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$ lies in the plane containing $L_1$ and $L_2$, and passes through the point of intersection of $L_1$ and $L_2$. If the line $L$ bisects the acute angle between the lines $L_1$ and $L_2$, then which of the following statements is/are TRUE?

Let ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ denotes the lines, $\overrightarrow{\mathrm{r}}=\widehat{\mathrm{i}}+\mathrm{\lambda }\left(-\widehat{\mathrm{i}}+2\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\lambda }\in \mathrm{R}$ and $\overrightarrow{\mathrm{r}}=\mathrm{\mu }\left(2\widehat{\mathrm{i}}-\widehat{\mathrm{j}}+2\widehat{\mathrm{k}}\right),\mathrm{\mu }\in \mathrm{R}$, respectively. If ${\mathrm{L}}_3$ is a line which is perpendicular to both ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ and cuts both of them, then which of the following options describe(s) ${\mathrm{L}}_3$ ?

Three lines are given by

$\overrightarrow{r}=\lambda \hat{i}, \lambda \in R$,

$\overrightarrow{r}=\mu \left(\widehat{i}+\widehat{j}\right),\mu \in R$ and

$\overrightarrow{r}=\nu \left(\widehat{i}+\widehat{j}+\widehat{k}\right),\nu \in R.$

Let the lines cut the plane $x+y+z=1$ at the points $A, B$ and $C$ respectively. If the area of the triangle $ABC$ is $\mathrm{\Delta }$ then the value of ${\left(6\mathrm{\Delta }\right)}^2$ equals_________

Let $P_1: 2x + y - z = 3 \mathrm{a}\mathrm{n}\mathrm{d} P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?