3D Geometry

Let $\mathrm{A}(x, y, z)$ be a point in $x y$-plane, which is equidistant from three points $(0,3,2),(2,0,3)$ and $(0,0,1)$.
Let $\mathrm{B}=(1,4,-1)$ and $\mathrm{C}=(2,0,-2)$. Then among the statements
(S1) : $\triangle \mathrm{ABC}$ is a...

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

The square of the distance of the point $ \left( \frac{15}{7}, \frac{32}{7}, 7 \right) $ from the line $ \frac{x + 1}{3} = \frac{y + 3}{5} = \frac{z + 5}{7} $ in the direction of the vector $ \hat{i} + 4\hat{j} + 7\hat{k} $ is:

Let the line passing through the points $(-1,2,1)$ and parallel to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ intersect the line $\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1}$ at the point $P$. Then the distance of $P$ from the point $...

Let in a $\triangle A B C$, the length of the side $A C$ be 6 , the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle A B C$ is:

Let P be the image of the point $\mathrm{Q}(7,-2,5)$ in the line $\mathrm{L}: \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $\mathrm{R}(5, \mathrm{p}, \mathrm{q})$ be a point on $L$. Then the square of the area of $\triangle P Q R$ is ______...

Let P be the foot of the perpendicular from the point $\mathrm{Q}(10,-3,-1)$ on the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$. Then the area of the right angled triangle $P Q R$, where $R$ is the point $(3,-2,1)$, is

The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ is :

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m$, $n$ are coprime numbers, then $m+n$ is equal to :

Let $\mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\mathrm{L}_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $\mathrm{L}_1$ an...

Let $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $\mathrm{L}_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}, \alpha \in \mathbf{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from...

Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3 \hat{i}+2 \hat{j}+2 \hat{k}$. If the distance of the point Q from the point $\mathrm{R}(1,3,3)$ is 5 , then the square of the area of $\trian...

The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $\mathrm{P}(2,-10,1)$, is :

The shortest distance between the lines \(\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}\) and \(\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}\) is:

Let the line \(\mathrm{L}\) intersect the lines \(x-2=-y=z-1,2(x+1)=2(y-1)=z+1\) and be parallel to the line \(\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}\). Then which of the following points lies on \(\mathrm{L}\) ?

The square of the distance of the image of the point \((6,1,5)\) in the line \(\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}\), from the origin is __________.

Consider the line \(\mathrm{L}\) passing through the points \((1,2,3)\) and \((2,3,5)\). The distance of the point \(\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)\) from the line \(\mathrm{L}\) along the line $$\frac{3 x-11}{2}=\fra...

If the shortest distance between the lines
$$\begin{array}{ll}
L_1: \vec{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, & \lambda \in \mathbb{R} \\
L_2: \vec{r}=2(1+\mu) \hat{i}+3(1+\mu) \hat{j}+(5+\mu) \hat{k}, & \mu \...

Let \(P(x, y, z)\) be a point in the first octant, whose projection in the \(x y\)-plane is the point \(Q\). Let \(O P=\gamma\); the angle between \(O Q\) and the positive \(x\)-axis be \(\theta\); and the angle between \(O P\) and the posi...

Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(1,6,4)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\). Then \(2 \alpha+\beta+\gamma\) is equal to ________

If the shortest distance between the lines \(\frac{x-\lambda}{2}=\frac{y-4}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8}\) is \(\frac{13}{\sqrt{29}}\), then a value of \(\lambda\) is :

Let \(P\) be the point \((10,-2,-1)\) and \(Q\) be the foot of the perpendicular drawn from the point \(R(1,7,6)\) on the line passing through the points \((2,-5,11)\) and \((-6,7,-5)\). Then the length of the line segment \(P Q\) is equal ...

The shortest distance between the lines \(\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}\) and \(\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}\) is

If \(A(3,1,-1), B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), C(2,2,1)\) and \(D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)\) are the vertices of a quadrilateral \(A B C D\), then its area is

If the shortest distance between the lines \(\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}\) and \(\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}\) is \(\frac{44}{\sqrt{30}}\), then the largest possible value of \(|\lambda|\) is equal to ___...

Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(3,-3,1)\) in the line \(\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}\) and \(\mathrm{R}\) be the point \((2,5,-1)\). If the area of the triangle \(\mathrm{PQR}\)...

If the line \(\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z\) makes a right angle with the line \(\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}\), then \(4 \lambda+9 \mu\) is equal to :

Let \(\mathrm{d}\) be the distance of the point of intersection of the lines \(\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}\) and \(\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}\) from the point \((7,8,9)\). Then \(\mathrm{d}^2+6\) is equal to :

Let the point \((-1, \alpha, \beta)\) lie on the line of the shortest distance between the lines \(\frac{x+2}{-3}=\frac{y-2}{4}=\frac{z-5}{2}\) and \(\frac{x+2}{-1}=\frac{y+6}{2}=\frac{z-1}{0}\). Then \((\alpha-\beta)^2\) is equal to ______...

Let \((\alpha, \beta, \gamma)\) be the image of the point \((8,5,7)\) in the line \(\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}\). Then \(\alpha+\beta+\gamma\) is equal to :

Let the point, on the line passing through the points \(P(1,-2,3)\) and \(Q(5,-4,7)\), farther from the origin and at a distance of 9 units from the point \(P\), be \((\alpha, \beta, \gamma)\). Then \(\alpha^2+\beta^2+\gamma^2\) is equal to...

Consider a line \(\mathrm{L}\) passing through the points \(\mathrm{P}(1,2,1)\) and \(\mathrm{Q}(2,1,-1)\). If the mirror image of the point \(\mathrm{A}(2,2,2)\) in the line \(\mathrm{L}\) is \((\alpha, \beta, \gamma)\), then $$\alpha+\bet...

Let \(\mathrm{P}\) be the point of intersection of the lines \(\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}\) and \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}\). Then, the shortest distance of \(\mathrm{P}\) from the line \(4 x=2 y=z\) is

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be the feet of perpendiculars from the point \(\mathrm{P}(a, a, a)\) on the lines \(x=y, z=1\) and \(x=-y, z=-1\) respectively. If \(\angle \mathrm{QPR}\) is a right angle, then \(12 a^2\) is equal to _...

A line passes through \(A(4,-6,-2)\) and \(B(16,-2,4)\). The point \(P(a, b, c)\), where \(a, b, c\) are non-negative integers, on the line \(A B\) lies at a distance of 21 units, from the point \(A\). The distance between the points $$P(a,...

The shortest distance, between lines \(L_1\) and \(L_2\), where \(L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}\) and \(L_2\) is the line, passing through the points \(\mathrm{A}(-4,4,3), \mathrm{B}(-1,6,3)\) and perpendicular to the line...

Let \((\alpha, \beta, \gamma)\) be the mirror image of the point \((2,3,5)\) in the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\). Then, \(2 \alpha+3 \beta+4 \gamma\) is equal to

If \(\mathrm{d}_1\) is the shortest distance between the lines \(x+1=2 y=-12 z, x=y+2=6 z-6\) and \(\mathrm{d}_2\) is the shortest distance between the lines $$\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}, \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-...

Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\). If the diagonal \(\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k}\), then the area of the parallelogram is equal to :

Let \((\alpha, \beta, \gamma)\) be the foot of perpendicular from the point \((1,2,3)\) on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\). Then \(19(\alpha+\beta+\gamma)\) is equal to :

Let a line passing through the point \((-1,2,3)\) intersect the lines \(L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}\) at \(M(\alpha, \beta, \gamma)\) and \(L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}\) at \(N(a, b, c)\). Then, the ...

Let \(L_1: \vec{r}=(\hat{i}-\hat{j}+2 \hat{k})+\lambda(\hat{i}-\hat{j}+2 \hat{k}), \lambda \in \mathbb{R}\),
$$L_2: \vec{r}=(\hat{j}-\hat{k})+\mu(3 \hat{i}+\hat{j}+p \hat{k}), \mu \in \mathbb{R} \text {, and } L_3: \vec{r}=\delta(\ell \hat{...

A line with direction ratios \(2,1,2\) meets the lines \(x=y+2=z\) and \(x+2=2 y=2 z\) respectively at the points \(\mathrm{P}\) and \(\mathrm{Q}\). If the length of the perpendicular from the point \((1,2,12)\) to the line \(\mathrm{PQ}\) ...

Let \(P Q R\) be a triangle with \(R(-1,4,2)\). Suppose \(M(2,1,2)\) is the mid point of \(\mathrm{PQ}\). The distance of the centroid of \(\triangle \mathrm{PQR}\) from the point of intersection of the lines $$\frac{x-2}{0}=\frac{y}{2}=\fr...

Let \(O\) be the origin and the position vectors of \(A\) and \(B\) be \(2 \hat{i}+2 \hat{j}+\hat{k}\) and \(2 \hat{i}+4 \hat{j}+4 \hat{k}\) respectively. If the internal bisector of \(\angle \mathrm{AOB}\) meets the line \(\mathrm{AB}\) at...

Let O be the origin, and M and \(\mathrm{N}\) be the points on the lines \(\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}\) and \(\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}\) respectively such that \(\mathrm{MN}\) is the shortest distance betwe...

Let \(\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)\) and \(\mathrm{R}(7,3,2)\) be the vertices of \(\triangle \mathrm{PQR}\). Then, the angle \(\angle \mathrm{QPR}\) is

If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum of all possible values of $\lambda$ is :

The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

The lines \(\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}\) and \(\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}\) intersect at the point \(P\). If the distance of \(\mathrm{P}\) from the line \(\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}\) is \(l\), ...