Angle Bisectors of Two Planes

Q. A plane passing through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$, also passes through the point:

1. ($\sqrt{2},1,4)$
2. ($\sqrt{2},-1,4)$
3. $(-\sqrt{2},-1,-4)$
4. ($-\sqrt{2},1,-4)$

Q.

The equation of the plane through (1, 2, 3) and parallel to the plane 2x + 3y - 4z = 0 is

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

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

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

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

Q. The equatio of the plane which bisects the angle between the planes 3x-6y+2z+5=0 and 4x-12y+3z-3=0 which contains the origin is

1. x-3y+z-5=0
2. 33x+13y+32z+45=0
3. None of these
4. 33x-13y+32z+45=0

Q. A plane which bisects the angle between the two given planes $2x–y+2z–4=0$ and $x+2y+2z–2=0,$ passes through the point :

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

Q. The equation of the plane passing through the line of intersection of the planes $\overrightarrow{r}\cdot (\hat{i}+\hat{j}+\hat{k})=1$ and $\overrightarrow{r}\cdot (2\hat{i}+3\hat{j}-\hat{k})+4=0$ and parallel to the $x-$axis is

1. $\overrightarrow{r}\cdot (\hat{i}+3\hat{k})+6=0$
2. $\overrightarrow{r}\cdot (\hat{j}-3\hat{k})+6=0$
3. $\overrightarrow{r}\cdot (\hat{j}-3\hat{k})-6=0$
4. $\overrightarrow{r}\cdot (\hat{i}-3\hat{k})+6=0$

Q. The equation of obtuse angular bisector of the planes $x-2y+2z+3=0,3x-6y-2z+2=0$ is

1. $2x-4y+8z-5=0$
2. $2x-4y-20z-15=0$
3. $12x-14y+z-11=0$
4. $16x-32y+8z+27=0$

Q. The equation of a plane bisecting the angle between the plane $2x-y+2z+3=0$ and $3x-2y+6z+8=0$ is/are:

1. $5x-y-4z-3=0$
2. $5x-y-4z-45=0$
3. $23x-13y+32z+45=0$
4. $23x-13y+32z+5=0$

Q. In $R^3$, let $P_1$ be the plane which contains the line $L_1:\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$ and $P_2$ be the another plane which contains the line $L_1$ and a point with position vector $\hat{j}$. If the vector $(\hat{i}+\hat{j})$ is normal to $P_1$, then which of the following is (are) true?

1. The equation of $P_1$ is $x+y=2$
2. The equation of $P_2$ is $\overrightarrow{r}\cdot (\hat{i}-2\hat{j}+\hat{k})=2$
3. The acute angle between $P_1$ and $P_2$ is ${\cot }^{-1}\left(\sqrt{3}\right)$
4. The angle between the plane $P_2$ and the line $L_1$ is ${\tan }^{-1}\left(\sqrt{3}\right)$

Q. Find the vector equation of the line passing through the point whose position vector is $3\hat{i}+\hat{j}-\hat{k}$ and which is parallel to the vector $2\hat{i}-\hat{j}+2\hat{k}$. If $P$ is a point on this line such that $AP=15$, find the position vector of $P$.

Q. The number of possible straight lines, passing through $(2,3)$ and forming a triangle with coordinate axes, whose area is $12sq.units$, is

1. $One$
2. $Two$
3. $Three$
4. $Four$

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

Q. Statement 1: A plane passes through the point A(2, 1, -3). If distance of this plane from origin is maximum, then its equation is $2x+y-3z=14$.
Statement 2: If the plane passing through the point $A\left(\overrightarrow{a}\right)$ is at maximum distance from origin, then normal to the plane is vector $\overrightarrow{a}$.

1. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
2. Statement 1 is false and Statement 2 is true.
3. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
4. Statement 1 is true and Statement 2 is false.

Q. The angle between plane

1. ${\sin }^{-1}\left(\frac{5}{\sqrt{33}}\right)$
2. ${\cos }^{-1}\left(\frac{5}{\sqrt{33}}\right)$
3. ${\sin }^{-1}\left(\frac{1}{\sqrt{33}}\right)$
4. ${\cos }^{-1}\left(\frac{3}{\sqrt{33}}\right)$

Q.

An equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the origin is

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

2. x - 2y + 2z + 1 = 0

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

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

Q. Find the equation of the plane 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. Find the equation of the plane passing through the intersection of the planes :
$x+y+z+1=0$ and $2x-3y+5z-2=0$ and the point $(-1,2,1)$.

Q. Find the equation of the plane passing through the intersection of the planes $2x+3y-z+1=0$ and $x+y-2z+3=0$, and perpendicular to the plane $3x-y-2z-4=0$.

Q. The equatio of the plane which bisects the angle between the planes 3x-6y+2z+5=0 and 4x-12y+3z-3=0 which contains the origin is

1. 33x-13y+32z+45=0
2. x-3y+z-5=0
3. 33x+13y+32z+45=0
4. None of these

Q. The domain of $f\left(x\right)=\sqrt{25-x^2}$ is

1. $(-\infty ,-5)$
2. $(5,\infty )$
3. $[-5,5]$
4. $[-\infty ,\infty ]$

Q.

An equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the origin is

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

2. x - 2y + 2z + 1 = 0

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

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

Q. Equation of the line which passes through the point with p.v. (2, 1, 0) and perpendicular to the plane containing the vectors $\hat{i}+\hat{j}and\hat{j}+\hat{k}$ is

1. $\overrightarrow{r}=(2,1,0)+t(1,-1,1)$
2. $\overrightarrow{r}=(2,1,0)+t(1,1,-1)$
3. $\overrightarrow{r}=(2,1,0)+t(-1,1,1)$
4. $\overrightarrow{r}=(2,1,0)+t(1,1,1)$

Q. In $R^3$, let $P_1$ be the plane which contains the line $L_1:\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$ and $P_2$ be the another plane which contains the line $L_1$ and a point with position vector $\hat{j}$. If the vector $(\hat{i}+\hat{j})$ is normal to $P_1$, then which of the following is (are) true?

1. The equation of $P_1$ is $x+y=2$
2. The equation of $P_2$ is $\overrightarrow{r}\cdot (\hat{i}-2\hat{j}+\hat{k})=2$
3. The acute angle between $P_1$ and $P_2$ is ${\cot }^{-1}\left(\sqrt{3}\right)$
4. The angle between the plane $P_2$ and the line $L_1$ is ${\tan }^{-1}\left(\sqrt{3}\right)$