3D Geometry

Let the image of the point \((1,0,7)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) be the point \((\alpha, \beta, \gamma)\). Then which one of the following points lies on the line passing through \((\alpha, \beta, \gamma)\) and...

Let the line of the shortest distance between the lines
$$
\begin{aligned}
& \mathrm{L}_1: \overrightarrow{\mathrm{r}}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k}) \text { and } \\\\
& \mathrm{L}_2: \overrightarrow{\mathrm...

If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all possible values of $\lambda$ is :

If the mirror image of the point $P(3,4,9)$ in the line
$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then 14 $(\alpha+\beta+\gamma)$ is :

Let $\mathrm{P}$ and $\mathrm{Q}$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the point $\mathrm{R}(1,2,3)$. If the centroid of the triangle PQR is $(\alpha, \beta, \gamma)$,...

Consider a $\triangle A B C$ where $A(1,3,2), B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle B A C$ meets
the line $B C$ at $D$, then the length of the projection of the vector $\overrightarrow{A D}$ on the vector $\overrightar...

Let \(\lambda_{1}, \lambda_{2}\) be the values of \(\lambda\) for which the points \(\left(\frac{5}{2}, 1, \lambda\right)\) and \((-2,0,1)\) are at equal distance from the plane \(2 x+3 y-6 z+7=0\). If \(\lambda_{1} > \lambda_{2}\), then th...

If the equation of the plane containing the line \(x+2 y+3 z-4=0=2 x+y-z+5\) and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2 \hat{j}+3 \hat{k})$ is $a x+b y+c z=4$, then \((a-b+c)\) i...

The shortest distance between the lines \(\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}\) and \(\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}\) is :

Let \(\mathrm{P}_{1}\) be the plane \(3 x-y-7 z=11\) and \(\mathrm{P}_{2}\) be the plane passing through the points \((2,-1,0),(2,0,-1)\), and \((5,1,1)\). If the foot of the perpendicular drawn from the point \((7,4,-1)\) on the line of in...

Let \(\mathrm{P}\) be the plane passing through the line \(\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}\) and the point \((2,4,-3)\). If the image of the point \((-1,3,4)\) in the plane P is \((\alpha, \beta, \gamma)\) then $$\alpha+\beta+\ga...

For \(\mathrm{a}, \mathrm{b} \in \mathbb{Z}\) and \(|\mathrm{a}-\mathrm{b}| \leq 10\), let the angle between the plane \(\mathrm{P}: \mathrm{ax}+y-\mathrm{z}=\mathrm{b}\) and the line \(l: x-1=\mathrm{a}-y=z+1\) be $$\cos ^{-1}\left(\frac{1...

Let the image of the point \(\mathrm{P}(1,2,3)\) in the plane \(2 x-y+z=9\) be \(\mathrm{Q}\). If the coordinates of the point \(\mathrm{R}\) are \((6,10,7)\), then the square of the area of the triangle \(\mathrm{PQR}\) is _____________.

One vertex of a rectangular parallelopiped is at the origin \(\mathrm{O}\) and the lengths of its edges along \(x, y\) and \(z\) axes are \(3,4\) and \(5\) units respectively. Let \(\mathrm{P}\) be the vertex \((3,4,5)\). Then the shortest ...

If the equation of the plane passing through the line of intersection of the planes \(2 x-y+z=3,4 x-3 y+5 z+9=0\) and parallel to the line \(\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}\) is \(a x+b y+c z+6=0\), then \(a+b+c\) is equal to :

If the lines \(\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}\) intersect, then the magnitude of the minimum value of \(8 \alpha \beta\) is _____________.

Let the line \(\mathrm{L}\) pass through the point \((0,1,2)\), intersect the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and be parallel to the plane \(2 x+y-3 z=4\). Then the distance of the point \(\mathrm{P}(1,-9,2)\) from the li...

A plane P contains the line of intersection of the plane \(\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=6\) and \(\vec{r} \cdot(2 \hat{i}+3 \hat{j}+4 \hat{k})=-5\). If \(\mathrm{P}\) passes through the point \((0,2,-2)\), then the square of dista...

Let \(\theta\) be the angle between the planes \(P_{1}: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9\) and \(P_{2}: \vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15\). Let \(\mathrm{L}\) be the line that meets \(P_{2}\) at the point \((4,-2,5)\) a...

Let the line \(L: \frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-3}{1}\) intersect the plane \(2 x+y+3 z=16\) at the point
\(P\). Let the point \(Q\) be the foot of perpendicular from the point \(R(1,-1,-3)\) on the line \(L\). If \(\alpha\) is the ...

Let the shortest distance between the lines
\(L: \frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}, \lambda \geq 0\) and
\(L_{1}: x+1=y-1=4-z\) be \(2 \sqrt{6}\). If \((\alpha, \beta, \gamma)\) lies on \(L\),
then which of the follo...

If a point $\mathrm{P}(\alpha, \beta, \gamma)$ satisfying
$$\left( {\matrix{
\alpha & \beta & \gamma \cr

} } \right)\left( {\matrix{
2 & {10} & 8 \cr
9 & 3 & 8 \cr
8 & 4 & 8 \cr

} } \right) = \left( {\matrix{
0...

Let $P$ be the plane, passing through the point $(1,-1,-5)$ and perpendicular to the line joining the points $(4,1,-3)$ and $(2,4,3)$. Then the distance of $P$ from the point $(3,-2,2)$ is :

The foot of perpendicular from the origin $\mathrm{O}$ to a plane $\mathrm{P}$ which meets the co-ordinate axes at the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ is $(2, \mathrm{a}, 4), \mathrm{a} \in \mathrm{N}$. If the volume of the tetr...

Let the plane $\mathrm{P}: 8 x+\alpha_{1} y+\alpha_{2} z+12=0$ be parallel to the line $\mathrm{L}: \frac{x+2}{2}=\frac{y-3}{3}=\frac{z+4}{5}$. If the

intercept of $\mathrm{P}$ on the $y$-axis is 1 , then the distance between $\mathrm{P}$ ...

If \(\lambda_{1} < \lambda_{2}\) are two values of \(\lambda\) such that the angle between the planes \(P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7\) and
\(P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9\) is $$\sin ^{-1}\left(...

If the equation of the plane passing through the point \((1,1,2)\) and perpendicular to the line \(x-3 y+ 2 z-1=0=4 x-y+z\) is \(\mathrm{A} x+\mathrm{B} y+\mathrm{C} z=1\), then \(140(\mathrm{C}-\mathrm{B}+\mathrm{A})\) is equal to ________...

The line \(l_1\) passes through the point (2, 6, 2) and is perpendicular to the plane \(2x+y-2z=10\). Then the shortest distance between the line \(l_1\) and the line \(\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}\) is :

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3 y-2 z-2=0=x-y+2 z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3 \alpha^2$ is equal to :

If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is :

A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$...

Let the co-ordinates of one vertex of \(\Delta ABC\) be \(A(0,2,\alpha)\) and the other two vertices lie on the line \({{x + \alpha } \over 5} = {{y - 1} \over 2} = {{z + 4} \over 3}\). For \(\alpha \in \mathbb{Z}\), if the area of $$\Delta...

Let the equation of the plane P containing the line \(x+10=\frac{8-y}{2}=z\) be \(ax+by+3z=2(a+b)\) and the distance of the plane \(P\) from the point (1, 27, 7) be \(c\). Then \(a^2+b^2+c^2\) is equal to __________.

The shortest distance between the lines \({{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}\) and \({{x - 1} \over 2} = {{y - 2} \over 1} = {{z - 6} \over { - 3}}\) is :

If the lines \({{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over 1}\) and \({{x - a} \over 2} = {{y + 2} \over 3} = {{z - 3} \over 1}\) intersect at the point P, then the distance of the point P from the plane \(z = a\) is :

The plane \(2x-y+z=4\) intersects the line segment joining the points A (\(a,-2,4)\) and B (\(2,b,-3)\) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is \(\sqrt5\). If \(ab < 0\) and P is the point $$(a-b...

Let the equation of the plane passing through the line \(x - 2y - z - 5 = 0 = x + y + 3z - 5\) and parallel to the line \(x + y + 2z - 7 = 0 = 2x + 3y + z - 2\) be \(ax + by + cz = 65\). Then the distance of the point (a, b, c) from the pla...

Consider the lines \(L_1\) and \(L_2\) given by
\({L_1}:{{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 2} \over 2}\)
\({L_2}:{{x - 2} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}\).
A line \(L_3\) having direction ratios 1, \(-\)1, \(-\)...

The distance of the point P(4, 6, \(-\)2) from the line passing through the point (\(-\)3, 2, 3) and parallel to a line with direction ratios 3, 3, \(-\)1 is equal to :

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line \({{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}\) is \(\alpha\), then 28\(\alpha^2\) is equal to ____________.

The shortest distance between the lines \(x+1=2y=-12z\) and \(x=y+2=6z-6\) is :

The foot of perpendicular of the point (2, 0, 5) on the line \({{x + 1} \over 2} = {{y - 1} \over 5} = {{z + 1} \over { - 1}}\) is (\(\alpha,\beta,\gamma\)). Then, which of the following is NOT correct?

The shortest distance between the lines \({{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}\) and \({{x - 6} \over 3} = {{1 - y} \over 2} = {{z + 8} \over 0}\) is equal to ________

The distance of the point (\(-1,9,-16\)) from the plane \(2x+3y-z=5\) measured parallel to the line \({{x + 4} \over 3} = {{2 - y} \over 4} = {{z - 3} \over {12}}\) is :

The distance of the point (7, \(-\)3, \(-\)4) from the plane passing through the points (2, \(-\)3, 1), (\(-\)1, 1, \(-\)2) and (3, \(-\)4, 2) is :

If the shortest between the lines \({{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}\) and \({{x - \lambda } \over 3} = {{y - 2\sqrt 6 } \over 4} = {{z + 2\sqrt 6 } \over 5}\) is 6, then the square of sum of al...

Let the plane containing the line of intersection of the planes P1 : \(x+(\lambda+4)y+z=1\) and P2 : \(2x+y+z=2\) pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of the point (2\(\lambda,\lambda,-\lambda\)) from the plane...

If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes \(x+2y+z=0\) and \(3y-z=3\) is (\(\alpha,\beta,\gamma\)), then \(\alpha+\beta+\gamma\) is equal to :

Let the image of the point \(P(2,-1,3)\) in the plane \(x+2 y-z=0\) be \(Q\). Then the distance of the plane \(3 x+2 y+z+29=0\) from the point \(Q\) is :

The shortest distance between the lines \({{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}\) and \({{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}\) is :