Family of Planes

Q. ntProve that the planes bx-ay=n, cy-bz=L, az-cx=m intersect in a line if aL+bm+cn=0n

Q. Show that the lines $\overrightarrow{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda \left(\hat{i}+2\hat{j}+2\hat{k}\right)\mathrm{and}\overrightarrow{r}=5\hat{i}-2\hat{j}+\mu \left(3\hat{i}+2\hat{j}+6\hat{k}\right)$ are intersecting. Hence, find their point of intersection.

Q. Find the shortest distance between the following pairs of parallel lines whose equations are:
(i) $\overrightarrow{r}=\left(\hat{i}+2\hat{j}+3\hat{k}\right)+\lambda \left(\hat{i}-\hat{j}+\hat{k}\right)\mathrm{and}\overrightarrow{r}=\left(2\hat{i}-\hat{j}-\hat{k}\right)+\mu \left(-\hat{i}+\hat{j}-\hat{k}\right)$

(ii) $\overrightarrow{r}=\left(\hat{i}+\hat{j}\right)+\lambda \left(2\hat{i}-\hat{j}+\hat{k}\right)\mathrm{and}\overrightarrow{r}=\left(2\hat{i}+\hat{j}-\hat{k}\right)+\mu \left(4\hat{i}-2\hat{j}+2\hat{k}\right)$

Q. The equation of a plane passing through $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=1$ and $\overline{r}.(2\hat{i}+\hat{k})=2$ can be given by

1. 
2. 
3. 
4. 

Q. The equation of a plane passing through $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=1$ and $\overline{r}.(2\hat{i}+\hat{k})=2$ can be given by

1. $\overline{r}.\left(\right(1+2\lambda )\hat{i}+\hat{j}+(1+\lambda \left)\hat{k}\right)=1+2\lambda$
2. $\overline{r}.\left(\right(\lambda +2)\hat{i}+\lambda \hat{j}+(\lambda +1\left)\hat{k}\right)=\lambda +2$
3. $\overline{r}.\left(\right(3\lambda \hat{i}+2\lambda \hat{j}+2\lambda \hat{k})=3)=3\lambda$
4. $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=1$

Q. The equation of a plane passing through $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=1$ and $\overline{r}.(2\hat{i}+\hat{k})=2$ can be given by

1. $\overline{r}.\left(\right(1+2\lambda )\hat{i}+\hat{j}+(1+\lambda \left)\hat{k}\right)=1+2\lambda$
2. $\overline{r}.\left(\right(\lambda +2)\hat{i}+\lambda \hat{j}+(\lambda +1\left)\hat{k}\right)=\lambda +2$
3. $\overline{r}.\left(\right(3\lambda \hat{i}+2\lambda \hat{j}+2\lambda \hat{k})=3)=3\lambda$
4. $\overline{r}.(\hat{i}+\hat{j}+\hat{k})=1$

Q. 2 planes can intersect in which of the following ways

1. Point
2. Line
3. Plane
4. Do not intersect at all

Q. A plane passing through the intersection of 2 planes $\overline{r}.\overline{n_1}=d_1$ and $\overline{r}.\overline{n_2}=d_2$ can be given by
$\overline{r}.(\overline{n_1}+\lambda \overline{n_1})=d_1+\lambda d_2$

1. True
2. False

Q. A plane passing through the intersection of 2 planes $\overline{r}.\overline{n_1}=d_1$ and $\overline{r}.\overline{n_2}=d_2$ can be given by
$\overline{r}.(\overline{n_1}+\lambda \overline{n_1})=d_1+\lambda d_2$

1. True
2. False

Q. 2 planes can intersect in which of the following ways

1. Point
2. Line
3. Plane
4. Do not intersect at all

Q. A plane passing through the intersection of 2 planes $\overline{r}.\overline{n_1}=d_1$ and $\overline{r}.\overline{n_2}=d_2$ can be given by
$\overline{r}.(\overline{n_1}+\lambda \overline{n_1})=d_1+\lambda d_2$

1. True
2. False

Q. Consider the equation $az+b\overline{z}+c=0$, where $a,b,cϵZ$

If $|a|\ne |b|$ , then $z$ represents

1. Circle
2. Ellipse
3. Straight line
4. One point

Q. Prove that the line $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(3\hat{i}-\hat{j}\right)\mathrm{and}\overrightarrow{r}=\left(4\hat{i}-\hat{k}\right)+\mu \left(2\hat{i}+3\hat{k}\right)$ intersect and find their point of intersection.