Family of Planes Passing through the Intersection of Two Planes

Q.

The straight line whose sum of the intercepts on the axes is equal to half of the product of the intercepts passes through the point

1. $\left(1,1\right)$

2. $\left(2,2\right)$

3. $\left(3,3\right)$

4. $\left(4,4\right)$

Q. Let $p_1:x-2y+3z=5$ and $p_2:2x+3y+z+4=0$ be two planes. If $P$ is the foot of the perpendicular dropped from the origin $O$ to the line of intersection of the planes, then

1. $\overrightarrow{OP}\cdot ((\hat{i}-2\hat{j}+3\hat{k})\times (2\hat{i}+3\hat{j}+\hat{k}))=0$
2. if $Q$ is another point on the line of intersection of planes, then equation of the plane containing a triangle $OPQ$ is $14x+7y+17z=0.$
3. equation of the plane perpendicular to the line of intersection of the planes and passing through $(1,1,1)$ is $11x-5y-7z+1=0.$
4. If $N_1$ and $N_2$ are foot of the perpendicular from the origin $O$ to the planes $p_1$ and $p_2$ respectively, then $O{N}_1+O{N}_2$ is equal to $\frac{9}{\sqrt{14}}.$

Q. The vector equation of the plane passing through the intersection of the planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{r}\cdot (\hat{i}-2\hat{j})=-2,$ and the point $(1,0,2)$ is

1. $\overrightarrow{r}\cdot (\hat{i}-7\hat{j}+3\hat{k})=\frac{7}{3}$
2. $\overrightarrow{r}\cdot (\hat{i}+7\hat{j}+3\hat{k})=7$
3. $\overrightarrow{r}\cdot (3\hat{i}+7\hat{j}+3\hat{k})=7$
4. $\overrightarrow{r}\cdot (\hat{i}+7\hat{j}+3\hat{k})=\frac{7}{3}$

Q.

The planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a straight line, where a, b, c are non-zero constants. Then the value of $a^2+b^2+c^2+2abc$ is

1. -1

2. 2

3. 1

4. 0

Q. If the equation of the plane containing the lines $x-y-z-4=0,x+y+2z-4=0$ and parallel to the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$ is $x+Ay+Bz+C=0$, then the value of $|A+B+C|$ is

Q. If for some $\alpha$ and $\beta$ in $R$, the intersection of the following three planes
$x+4y-2z=1x+7y-5z=\beta x+5y+\alpha z=5$
is a line in $R^3$, then $\alpha +\beta$ is equal to :

1. $0$
2. $10$
3. $-10$
4. $2$

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

1. 2x + 3y + 4z = 9
2. x + y + z = 3
3. x + 2y + 3z = 6
4. 20x + 23y + 26z = 69

Q. The plane through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to $y$-axis also passes through the point:

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

Q. The plane through the intersection of the planes $x+y+z=1$ and $2x+3y-z+4=0$ and parallel to $y$-axis also passes through the point:

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

Q. The equation of the plane passing through the line of intersection of the planes 3x-y-4z=0 and x+3y+6=0 whose distance from the origin is 1, is

1. x -2y -2z-3 =0, 2x+y -2z+3 =0
2. x -2y+2z-3 =0, 2x+y+2z+3 =0
3. x+2y-2z-3=0, 2x-y-2z+3 =0
4. None of these

Q. A plane $P:ax+by+cz=1$ passes through the intersection of planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=-3$ and $\overrightarrow{r}\cdot (\hat{i}-\hat{j}+\hat{k})=2.$ If plane $P$ divides the line segment joining $M(3,0,2)$ and $N(0,3,-1)$ in the ratio $2:1$ internally, then $(a+b+c)$ is equal to

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

1. 2x + 3y + 4z = 9
2. x + y + z = 3
3. x + 2y + 3z = 6
4. 20x + 23y + 26z = 69

Q. The plane $P_1:4x+7y+4z+81=0$ is rotated through a right angle about its line of intersection with the plane $P_2:5x+3y+10z=25.$ If the plane in its new position be denoted by $P$ and the distance of plane $P$ from the origin is $d$ units, then the value of $\left[d/2\right],$ where $[.]$ represents the greatest integer function, is