Equation of Planes Parallel to Axes

Q. The equation of the plane through the intersection of the planes x+y+z=1 and 2x+3y-z+4=0 parallel to x -axis is

1. y-3z+6=0
2. 3y-z+6=0
3. y+3z+6=0
4. 3y-2z+6=0

Q. Write the equation of the line through the points $(1,–1)\text{and}(3,5)$.

Q. The equation of a plane parallel to x - axis is

1. ax+by+d=0
2. ax+cz+d=0
3. ax+by+cz+d=0
4. by+cz+d=0

Q. The equation of the plane parallel to the plane $2x+3y+4z+5=0$ and passing through the point $(1,1,1)$ is:

1. $2x+3y+4z+9=0$
2. $2x+3y+4z-9=0$
3. $2x+3y+4z+7=0$
4. $2x+3y+4z-7=0$

Q.

Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.

Q. A plane $\pi$ makes intercept 3 and 4 respectively on z - axis and x - axis. If $\pi$ is parallel to y - axis, then its equation is

1. 3x+4z=12
2. 3z+4y=12
3. 3y+4z=12
4. 3z+4x=12

Q. If the plane $3x-4y+5z=0$ is parallel to $\frac{2x-1}{2}=\frac{1-y}{3}=\frac{z-2}{a},$ then the value of $a$ is:

1. $\frac{6}{4}$
2. $-3$
3. $\frac{1}{4}$
4. $\frac{3}{4}$

Q. If the equation of the plane through the points $(2,2,1)$ and $(9,3,6)$ and perpendicular to the plane $2x+6y+6z=9$ is $px+qy+rz=9,$ then $p+q+r=$

Q.

The equation of the plane containing the line of intersection of the planes 2x - y = 0 and y - 3z = 0 and perpendicular to the plane 4x + 5y - 3z - 8 = 0 is

1. 
   

2. 
   

3. 
   

4. 
   

Q. Find the image of the point having position vector $\hat{i}+3\hat{j}+4\hat{k}$ in the plane$\overrightarrow{r}.(2\hat{i}–\hat{j}+\hat{k})+3=0$.

Q. Consider a plane $x+y-z=1$ and point $A(1,2,-3).$ A line $L$ has the equation $x=1+3r,y=2-r$ and $z=3+4r.$

The coordinate of a point $B$ of Line $L$ such that $AB$ is parallel to the plane is

1. $(10,-1,15)$
2. $(-5,4,-5)$
3. $(4,1,7)$
4. $(-8,5,-9)$

Q. The plane passing through the points $(1,2,1),$ $(2,1,2)$ and parallel to the line, $2x=3y,z=1$ also passes through the point:

1. $(0,-6,2)$
2. $(0,6,-2)$
3. $(-2,0,1)$
4. $(2,0,-1)$

Q. Consider the lines
$L_1:x=3-t,y=2+t,z=5t$ intersecting the plane $x-y+2z=9$ at the point $A$
$L_2:x=1+2t,y=4t,z=2-3t$ intersecting the plane $x+2y-z+1=0$ at the point $B$ and
$L_3:x=y-1=2z$ intersecting the plane $4x-y+3z=8$ at the point $C$. Then the points $A,B,C$

1. do not form a triangle.
2. constitute an obtuse triangle.
3. constitute an acute triangle.
4. constitute a right triangle.

Q. A plane bisects the line segment joining the points $(1,2,3)$ and $(-3,4,5)$ at right angles. Then this plane also passes through the point :

1. $(-1,2,3)$
2. $(1,2,-3)$
3. $(3,2,1)$
4. $(-3,2,1)$

Q. The equation of plane passes through $(1,2,3)$ and perpendicular to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{1}$ is

1. $x+2y+3z=14$
2. $x-2y-3z+12=0$
3. $3x+2y+z=8$
4. $2x+3y+z=11$

Q. Find the equation of the line which is passing through the points $(-1,1)$ and $(2,-4)$.

Q.

The equation of the plane through the intersection of the plane $3x-y+2z=0$, $x+y+z-2=0$ and the point $(2,2,1)$ is

1. $7x+5y+4z+8=0$

2. $7x+5y+4z-8=0$

3. $7x-5y+4z-8=0$

4. None of these

Q. Projection of the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ on the plane $3x-6y+3z=108$ is

1. $\frac{x}{1}=\frac{y+24}{2}=\frac{z+12}{3}$
2. $\frac{x-3}{3}=\frac{y+6}{-6}=\frac{z-3}{3}$
3. $\frac{x-6}{1}=\frac{y+12}{2}=\frac{z-6}{3}$
4. $\frac{x-6}{3}=\frac{y+12}{-6}=\frac{z-6}{3}$

Q. 16. Convert this equation of plane to vector form 5x+7y+9z+4=0

Q.

Find the equation of the plane through the line of intersection of the planes x+y+z=1 and 2x+3y+4z=5 which is perpendicular to the plane x-y+z=0.

Q. The equation of the plane, which bisects the line joining the points $(1,2,3)$ and $(3,4,5)$ at right angles is,

1. $x+y+z=0$
2. $x+y-z=9$
3. $x+y+z=9$
4. $x+y-z+9=0$

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

1. $2x+y+2=0$
2. $3x+y-z=2$
3. $2x-3y+8z=3$
4. $4x+3y+4=0$

Q. The plane passing through the point $(-2,-2-3)$ and containing the line through the points $(1,1,1)$ and $(1,-1,2)$ make intercepts $a,b,c$ along the $x,y,z$ axis respectively. Then the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ is:

1. $3$
2. $4$
3. $6$
4. $1$

Q. The point $P$ is the intersection of the straight line joining the points $Q(2,3,5)$ and $R(1,–1,4)$ with the plane $5x–4y–z=1$. If $S$ is the foot of the perpendicular drawn from the point $T(2,1,4)$ to $QR$, then the length of the line segment $PS$ is

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

Q. Find the intercepts cut off by the plane $2x+y-z=5$.

Q. The equation of a plane passing through the point $A(2,-3,7)$ and making equal intercepts on the axes, is?

1. $x+y+z=6$
2. $x+y+z=9$
3. $x+y+z=4$
4. $x+y+z=3$

Q. The equation of a plane parallel to x - axis is

1. ax+by+d=0
2. by+cz+d=0
3. ax+cz+d=0
4. ax+by+cz+d=0

Q. If the line $\frac{x-{\lambda }^2}{-3\lambda }=\frac{y+5}{\lambda }=\frac{z+3}{-1}$ lies on the plane $x+\lambda y-2z=0,$ then the value of $\lambda$ is equal to

1. $1$
2. $2$
3. $3$
4. $4$

Q. The equation of a plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=$ $\frac{z-3}{2}$, and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at greatest distance from point $(0,0,0)$ is

1. $4x+3y+5z=25$
2. $4x+3y+5z=50$
3. $3x+4y+5z=49$
4. $x+7y-5z=2$

Q. A plane $\pi$ makes intercept 3 and 4 respectively on z - axis and x - axis. If $\pi$ is parallel to y - axis, then its equation is

1. 3y+4z=12
2. 3z+4y=12
3. 3x+4z=12
4. 3z+4x=12