Three-Dimensional Geometry

Let L$_1$ be a line in the yz - plane with y and z intercepts $\dfrac14$ and $\dfrac1C$ (C > 0) respectively. Let L$_2$ be the line in the xz - plane with x and z intercepts $\dfrac13$ and $-\dfrac1C$ respectively. If the shortest distance between L$_1$ and L$_2$ is $\dfrac15$, then C$^2$ is equal to _______.

The value of $D$ is ____.

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

The distance of the point (1, 1, 9) from the point of intersection of the line $\dfrac{x-3}1=\dfrac{y-4}2=\dfrac{z-5}2$ and the plane $x+y+z=17$ is :

Let $L_1$ and $L_2$ be the following straight lines.

$$L_1: \frac{x-1}{1} = \frac{y}{-1} = \frac{z-1}{3}\quad\text{and}\quad L_2: \frac{x-1}{-3} = \frac{y}{-1} = \frac{z-1}{1}.$$

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

Let $L_1$ and $L_2$ denote the lines

$$\vec r = \hat i + \lambda(-\hat i + 2\hat j + 2\hat k),\ \lambda\in\mathbb R\quad\text{and}\quad\vec r = \mu(2\hat i - \hat j + 2\hat k),\ \mu\in\mathbb R$$

respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

Three lines are given by

$$\vec r = \lambda\hat i,\ \lambda\in\mathbb R;\quad\vec r = \mu(\hat i + \hat j),\ \mu\in\mathbb R\quad\text{and}\quad\vec r = v(\hat i + \hat j + \hat k),\ v\in\mathbb R.$$

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 $\Delta$ then the value of $(6\Delta)^2$ equals ______

Three lines

$$L_1:\ \vec r = \lambda\hat i,\ \lambda\in\mathbb R,$$

$$L_2:\ \vec r = \hat k + \mu\hat j,\ \mu\in\mathbb R\quad\text{and}$$

$$L_3:\ \vec r = \hat i + \hat j + v\hat k,\ v\in\mathbb R$$

are given. For which point(s) $Q$ on $L_2$ 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 $P_1: 2x + y - z = 3$ and $P_2: x + 2y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?

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 $(1, 1, 1)$ and perpendicular to the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$, is

The distance of the point (1, 3, −7) from the plane passing through the point (1, −1, −1), having normal perpendicular to both the lines $\dfrac{x-1}1=\dfrac{y+2}{-2}=\dfrac{z-4}3$ and $\dfrac{x-2}2=\dfrac{y+1}{-1}=\dfrac{z+7}{-1}$, is :

If the image of the point P(1, −2, 3) in the plane, $2x+3y-4z+22=0$ measured parallel to the line, $\dfrac x1=\dfrac y4=\dfrac z5$ is Q, then PQ is equal to :

The coordinates of the foot of the perpendicular from the point (1, −2, 1) on the plane containing the lines, $\dfrac{x+1}6=\dfrac{y-1}7=\dfrac{z-3}8$ and $\dfrac{x-1}3=\dfrac{y-2}5=\dfrac{z-3}7$, is :

The line of intersection of the planes $\vec r\cdot(3\hat i-\hat j+\hat k)=1$ and $\vec r\cdot(\hat i+4\hat j-2\hat k)=2$, is :

If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of ΔABC is :

If the line, $\dfrac{x-3}1=\dfrac{y+2}{-1}=\dfrac{z+\lambda}{-2}$ lies in the plane, $2x-4y+3z=2$, then the shortest distance between this line and the line, $\dfrac{x-1}{12}=\dfrac y9=\dfrac z4$ is :

Consider a pyramid $OPQRS$ located in the first octant $(x\ge0,\ y\ge0,\ z\ge0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP=3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS=3$. Then

Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through $P$ and containing the straight line $\dfrac x1=\dfrac y2=\dfrac z1$ is