Equation of Plane Containing Two Lines

Q. Equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane $5x+3y-6z+8=0$ is

1. $33x+45y+50z=41$
2. $33x+45y+50z=-41$
3. $35x+45y+50z=41$
4. $35x+45y+50z=-41$

Q. The perpendicular distance from the origin to the plane containing the two lines,
$\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ , is:

1. $11$
2. $11\sqrt{6}$
3. $\frac{11}{\sqrt{6}}$
4. $6\sqrt{11}$

Q. The projection of the line $L:\frac{x+1}{-1}=\frac{y}{2}=\frac{z-1}{3}$ on the plane $\pi :x-2y+z=6$ is the line of intersection of this plane $\pi$ with the plane (containing the line $L$)

1. $2x+y+2=0$
2. $3x+y-z=2$
3. $2x-3y+8z=3$
4. none of these

Q. If two lines
$\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}$ and $\frac{x-{\alpha }^{\prime }}{l^{\prime }}=\frac{y-{\beta }^{\prime }}{m^{\prime }}=\frac{z-{\gamma }^{\prime }}{n^{\prime }}$
lie in the same plane then the equation of the plane is -

1. $\left|\begin{array}{ccc}x& y& z\\ l& m& n\\ l^{\prime }& m^{\prime }& n^{\prime }\end{array}\right|=0$
2. $\left|\begin{array}{ccc}x-\alpha & y-\beta & z-\gamma \\ l& m& n\\ l^{\prime }& m^{\prime }& n^{\prime }\end{array}\right|=0$
3. $\left|\begin{array}{ccc}l-l^{\prime }& m-m^{\prime }& n-n^{\prime }\\ l& m& n\\ l^{\prime }& m^{\prime }& n^{\prime }\end{array}\right|=0$
4. None of these

Q. The equation of the plane passing through the lines $\frac{x-4}{1}=\frac{y-3}{1}=\frac{z-2}{2}$ and $\frac{x-3}{1}=\frac{y-2}{-4}=\frac{z}{5}$ is

1. 11x-y-3z=35
2. 11x+y-3z=35
3. 11x-y+3z=35
4. None of these

Q. The projection of the line $L:\frac{x+1}{-1}=\frac{y}{2}=\frac{z-1}{3}$ on the plane $\pi :x-2y+z=6$ is the line of intersection of this plane $\pi$ with the plane (containing the line $L$)

1. $2x+y+2=0$
2. $3x+y-z=2$
3. $2x-3y+8z=3$
4. none of these

Q.

The equation of the plane containing the line 2x - 5y + z =3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z =1 is

1. 2x + 6y + 12z = 13

2. x + 3y + 6z = -7

3. x + 3y + 6z = 7

4. 2x + 6y + 12x = -13

Q. The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is :

1. $3$
2. $\sqrt{3}$
3. $\frac{1}{3}$
4. $\frac{1}{\sqrt{3}}$

Q. The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}\text{and}\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$is :

1. $x+2y-2z=0$
2. $3x+2y-3z=0$
3. $x-2y+z=0$
4. $5x+2y-4z=0$

Q.

Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the staight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

1. x +2y - 2z = 0

2. 3x + 2y - 2z = 0

3. x - 2y + z =0

4. 5x + 2y - 4z = 0