Condition for Two Lines to Parallel

Q. Find the equation of a line parallel to x -axis and passing through the origin.

Q. If any tangent plane to the sphere $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ makes intercepts $a,b,c$ with the coordinate axes at $A,B,C$ and $P$ be the center of the sphere. Then

1. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{4}{r^2}$
2. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{r^2}$
3. Volume of the tetrahedron $PABC=\frac{abc}{3}$
4. Volume of the tetrahedron $PABC=\frac{abc}{6}$

Q. The acute angle bisector between the intersecting lines $\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}$ and $\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$ is

1. $\frac{x-1}{-1}=\frac{y}{1}=\frac{z-1}{2}$
2. $\frac{x-1}{-1}=\frac{y}{-1}=\frac{z-1}{2}$
3. $\frac{x-1}{2}=\frac{z-1}{1},y=0$
4. $\frac{x-1}{-2}=\frac{y}{-1}=\frac{z-1}{1}$

Q. Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (−1, −2, 1), (1, 2, 5).

Q. If any tangent plane to the sphere $(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=r^2$ makes intercepts $a,b,c$ with the coordinate axes at $A,B,C$ and $P$ be the center of the sphere. Then

1. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{4}{r^2}$
2. $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{r^2}$
3. Volume of the tetrahedron $PABC=\frac{abc}{3}$
4. Volume of the tetrahedron $PABC=\frac{abc}{6}$

Q. The equation of line passing through point $A(2,-1,1)$ and parallel to vector $2\hat{i}+3\hat{j}-\hat{k}$ is

1. $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z-1}{-1}$
2. $\frac{x+2}{-2}=\frac{y+1}{-1}=\frac{z-1}{-1}$
3. $\frac{x-2}{2}=\frac{y-1}{3}=\frac{z-1}{-1}$
4. $\frac{x+2}{2}=\frac{y-1}{3}=\frac{z-1}{-1}$

Q. The vector equation of the line passing through the point $(-1,-1,2)$ and parallel to the line $2x-2=3y+1=6z-2,$ is

1. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (3\hat{i}+2\hat{j}+\hat{k})$
2. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (3\hat{i}+2\hat{j}-\hat{k})$
3. $\overrightarrow{r}=(\hat{i}+\hat{j}-2\hat{k})+\lambda (3\hat{i}+2\hat{j}+\hat{k})$
4. $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k})$