Equation of Plane through Origin

Q.

The equation of the lines represented by the equation $ab(x^2-y^2)+(a^2-b^2)xy=0$are

1. $ax–by=0,bx+ay=0$

2. $ax–by=0,bx–ay=0$

3. $ax+by=0,bx+ay=0$

4. $ax+by=0,bx–ay=0$

Q. The equation of the plane passing through the intersection of the planes 2x – 5y + z = 3 and x + y + 4z = 5 and Parallel to the plane x + 3y + 6z = 1 is x + 3y + 6z = k, where k is

1. 3
2. 7
3. 5
4. 2

Q.

Find the equation of the plane through the intersection of the planes 3x-y+2z-4=0 and x+y+z-2=0 and the point(2, 2, 1).

Q. The equation of the plane passing through the intersection of the planes x+2y+3z+4 = 0 and 4x+3y+2z+1=0 and the origin is

1. 3x+2y+z+1=0
2. 3x+2y+z=0
3. 2x+3y+z=0
4. x+y+z=0

Q. The equation of the plane passing through the intersection of the planes x+2y+3z+4 = 0 and 4x+3y+2z+1=0 and the origin is

1. 3x+2y+z+1=0
2. 3x+2y+z=0
3. 2x+3y+z=0
4. x+y+z=0

Q. Consider the following system of equations :
$ax+by+cz=0az+bx+cy=0ay+bz+cx=0$
$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If}a+b+c\ne 0\text{and}& \text{(P)}& \text{Planes meet only at one point}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0.& & \\ \text{(II)}& \text{If}a+b+c=0\text{and}& \left(Q\right)& \text{Equations represent the line}x=y=z\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(III)}& \text{If}a+b+c\ne 0\text{and}& \text{(R)}& \text{Equations represent identical planes}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(IV)}& \text{If}a+b+c=0\text{and}& \text{(S)}& \text{The solution of the system represents}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0& & \text{whole of the three dimensional space}\end{array}$


Which of the following is the "INCORRECT" option?

1. $\left(IV\right)\to \left(S\right)$
2. $\left(II\right)\to \left(Q\right)$
3. $\left(I\right)\to \left(S\right)$
4. $\left(III\right)\to \left(P\right)$

Q.

Choose the correct answer in Question
Distance between the two planes 2x + 3y + 4z = 4 and 4x + 6y+ 8z =12 is
(a) 2 units
(b) 4 units
(c) 8 units
(d) $\frac{2}{\sqrt{29}}units$

Q. Find the vector equation of the plane passing through the intersection of the planes
$\overrightarrow{r}.(2\hat{i}+2\hat{j}-3\hat{k})=7,\overrightarrow{r}.(2\hat{i}+5\hat{j}+3\hat{k})=9$ and through the point $(2,1,3)$

Q. Consider the following system of equations :
$ax+by+cz=0az+bx+cy=0ay+bz+cx=0$
$\begin{array}{cccc}& \text{List - I}& & \text{List - II}\\ \text{(I)}& \text{If}a+b+c\ne 0\text{and}& \text{(P)}& \text{Planes meet only at one point}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0.& & \\ \text{(II)}& \text{If}a+b+c=0\text{and}& \left(Q\right)& \text{Equations represent the line}x=y=z\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(III)}& \text{If}a+b+c\ne 0\text{and}& \text{(R)}& \text{Equations represent identical planes}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2\ne 0& & \\ \text{(IV)}& \text{If}a+b+c=0\text{and}& \text{(S)}& \text{The solution of the system represents}\\ & (a-b{)}^2+(b-c{)}^2+(c-a{)}^2=0& & \text{whole of the three dimensional space}\end{array}$


Which of the following is the "INCORRECT" option?

1. $\left(IV\right)\to \left(S\right)$
2. $\left(II\right)\to \left(Q\right)$
3. $\left(I\right)\to \left(S\right)$
4. $\left(III\right)\to \left(P\right)$

Q. Equation of the plane passing through the intersection of the planes $x+y+z=6$ and $2x+3y+4z+5=0$ and the point $(1,1,1)$ is

1. $20x+23y+26z-69=0$
2. $31x+45y+49z+52=0$
3. $8x+5y+2z-69=0$
4. $4x+5y+6z-7=0$

Q. The vector equation of the plane through the point $\hat{i}+2\hat{j}-\hat{k}$ and perpendicular to the line of intersection of the plane $r.(3\hat{i}-\hat{j}+\hat{k})=1$ and $r.(3\hat{i}-\hat{j}+\hat{k})=2$ is

1. $r.(2\hat{i}+7\hat{j}-13\hat{k})=1$
2. $r.(2\hat{i}-7\hat{j}-13\hat{k})=1$
3. $r.(2\hat{i}+7\hat{j}+13\hat{k})=0$
4. None of the above

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

1. $3x+y+\left(\frac{5}{3}\right)z=0$
2. $x+y+z=0$
3. $9x–3y+z=0$
4. $4x+y+2z=0$

Q. The equation of the plane through the intersection of the planes $x+2y+3z-4=0$ and $4x+3y+3z+1=0$ and passing through the origin is-

1. $x+14y+11z=0$
2. $7x+4y+z=0$
3. $17x+14y+15z=0$
4. $17x+y+z=0$

Q.

Which of the following is the graph of equation $f\left(x\right)=2\left|x-4\right|-3$

1. 

2. 

3. 

4. 

Q. If a plane passes through intersection of planes $2x-y-4=0$ and $y+2z-4=0$ and also passes through the point $(1,1,0)$. Then the equation of plane is

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