Condition for Two Lines to Be on the Same Plane

Q.

Two lines $L_1:x=5,\frac{y}{3-\alpha }=\frac{z}{-2}$ and $L_2:x=\alpha ,\frac{y}{-1}=\frac{z}{2-\alpha }$ are coplanar. Then, $\alpha$ can take value(s)

1. 1

2. 2

3. 3

4. 4

Q.

Find the equation of the plane containing the lines:

$\frac{(x-5)}{4}=\frac{(y-7)}{4}=\frac{(z+3)}{-5}$ and $\frac{(x-8)}{7}=\frac{(y-4)}{1}=\frac{(z-5)}{3}$.

Q.

The equation of the line joining the origin to the points of intersection of the curve $x^2+y^2=a^2$and $x^2+y^2-ax-ay=0$ is:

1. $x^2-y^2=0$

2. $xy=0$

3. $xy-x^2=0$

4. $y^2+xy=0$

Q. If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{1}=\frac{z}{1}$ intersect, then k=

1. $\frac{2}{9}$
2. $\frac{9}{2}$
3. $0$
4. None of these

Q. Find the shortest distance between the lines
$\overrightarrow{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda (\hat{i}-\hat{j}+\hat{k})$ and
$\overrightarrow{r}=\widehat{2i}-\hat{j}-\hat{k}+\mu (\widehat{2i}+\hat{j}+2\hat{k})$