Equation of a Plane : Point Normal Form

Q. The equation of the plane passing through the point (–2, –2, 2) and containing the line joining the points (1, 1, 1) and (1, –1, 2) is :

1. x + 2y – 3z + 4 = 0
2. 5x + 2y – 3z – 17 = 0
3. x – 3y – 6z + 8 = 0
4. 3z – 4y + 1 = 0

Q.

The equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + = 0, is

1. 
   

2. 
   

3. 
   

4. 
   

Q. If the point $(2,\alpha ,\beta )$ lies on the plane which passes through the point $(3,4,2)$ and$(7,0,6)$ and is perpendicular to the plane $2x-5y=15$, then $2\alpha -3\beta$ is equal to :

1. $17$
2. $12$
3. $5$
4. $7$

Q.

The equation of the plane passing through the origin and perpendicular to the line x=2 y=3 z is

1. 
   

2. 
   

3. None of these

4. 
   

Q. Let $P$ be the plane containing the line $L_1:y+z=2,$ $x=0$ and is parallel to the line $L_2:x-z=2,y=0.$ If the distance of the plane $P$ from the origin is $d$ units, then the value of $3d^2$ is

Q.

If the image of the point $P\left(1,-2,3\right)$ in the plane, $2x+3y-4z+22=0$ measured parallel to the line, $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is $Q$, then $PQ$ is equal to

1. $2\sqrt{42}$

2. $\sqrt{42}$

3. $6\sqrt{5}$

4. $3\sqrt{5}$

Q. A plane $P$ contains the line $L_1:\frac{y}{b}+\frac{z}{c}=1,x=0$ and is parallel to the line $L_2:\frac{x}{a}-\frac{z}{c}=1,y=0$, then which of the following is/are correct?

1. If the shortest distance between $L_1$ and $L_2$ is $\frac{1}{4}$, then value of $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$ is $16$
2. If the shortest distance between $L_1$ and $L_2$ is $\frac{1}{4}$ then value of $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}$ is $64$
3. Distance of image of $A(a,0,0)$ in the plane $P$ from $M(\frac{-5}{3},\frac{8}{3},\frac{11}{3}),$ where $a=b=c=1$ is equal to $2$
4. Distance of image of $A(a,0,0)$ in the plane $P$ from $M(\frac{-5}{3},\frac{8}{3},\frac{11}{3}),$ where $a=b=c=1$ is equal to $3$

Q. Equation of the plane perpendicular to the plane x – 2y + 5z + 1 = 0 which passes through the points (2, –3, 1) and (–1, 1, –7) is given by :

1. 4x – 4y + z + 7 = 0
2. 4x + 7y + 2z + 11 = 0
3. 2x + y – z = 0
4. 2x + y – 3z = 0

Q.

The distance of the point (1, 3, -7) from the plane passing through the point (1, -1, -1) having normal perpendicular to both the lines
$\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$, is

1. $\frac{20}{\sqrt{74}}$ units

2. $\frac{10}{\sqrt{83}}$ units

3. $\frac{5}{\sqrt{83}}$ units

4. $\frac{10}{\sqrt{74}}$ units

Q. The equation of the plane equally inclined to the lines $L_1:\frac{x-1}{2}=\frac{y}{-2}=\frac{z+2}{-1}$ and $L_2:\frac{x+3}{8}=\frac{y-4}{1}=\frac{z}{-4}$ and passing through the origin is/are

1. $14x-5y-7z=0$
2. $2x+7y-z=0$
3. $2x+7y+z=0$
4. $14x+5y-7z=0$

Q.

Find the equation of plane if it passes through a point (2, 3, - 4) and is perpendicular to the line with direction ratios (2, 3, -1) .

1. 3x + 2y - z = 17

2. 2x + 3y - z = 13

3. 3x + 2y - z = 13

4. 2x + 3y - z = 17

Q. Equation of the plane through the point (1, 0, –2) and perpendicular to the planes 2x + y – z – 2 = 0 and x – y –z – 3 = 0 is given by :

1. 6x – y + 3z = 0
2. 2x – y + 3z + 4 = 0
3. 2x + y – z + 4 = 0
4. 5x + y – z + 4 = 0

Q.

Find the equation of the plane through the points (2, 1, 0), (3, -2, -2) and (3, 1, 7).

Q. Let the projection of the line $L:\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z-2}{3}$ in the plane $3x+2y+z=5$ is $L^{\prime }$. If the distance of a plane from the origin which contains the lines $L$ and $L^{\prime }$ is $p$, then the value of $6p^2$ is

Q. A plane which contain the line $\frac{y}{b}+\frac{z}{c}=1,x=0$, and parallel to the line $\frac{x}{a}-\frac{z}{c}=1,y=0$. If $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are the direction cosines of the a line $l$ then distance of the plane from the origin is

Q. P is a fixed point (a, a, a) on a line through the origin equally inclined to the axes, then any plane through P perpendicular to OP, makes intercepts on the axes, the sum of whose reciprocals is equal to

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

Q. The equation to the plane through the points (2, -1, 0), (3, -4, 5) parallel to a line with direction cosines proportional to (2, 3, 4 ) is 9x – 2y – 3z = k, where k is

1. 20
2. -20
3. 10
4. -10

Q.

A plane P passes through a point P (3, -2, 1) and is perpendicular to the vector

$\overrightarrow{V}=4\hat{i}+7\hat{j}-4\hat{k}.$ The distance between the plane P and the plane. $\overrightarrow{r}.(4\hat{i}+7\hat{j}-4\hat{k})+33=0$

1. 3

2. 2

3. 1

4. $\frac{28}{9}$

Q. The equation of a plane passing through (1, 2, –3), (0, 0, 0) and perpendicular to the plane 3x – 5y + 2z = 11, is :

1. 4x + y + 2z = 0
2. x + y + z = 0
3. $3x+y+\frac{5}{3}z=0$
4. $3x-y+\frac{z}{3}=0$

Q. The vector equation of the line passing through $3\hat{i}-5\hat{j}+7\hat{k}$ and perpendicular to the plane $3x-4y+5z=8$ is
(where $\lambda$ is a parameter)

1. $3\hat{i}-5\hat{j}+7\hat{k}+\lambda (3\hat{i}+5\hat{j}-4\hat{k})$
2. $3\hat{i}-5\hat{j}+7\hat{k}+\lambda (2\hat{i}-\hat{j}+3\hat{k})$
3. $3\hat{i}-5\hat{j}+7\hat{k}+\lambda (3\hat{i}-4\hat{j}+5\hat{k})$
4. $3\hat{i}-5\hat{j}+7\hat{k}+\lambda (2\hat{i}+3\hat{j}+5\hat{k})$

Q.

Find a range of values of $c$ such that the lines $x-y=2$ and $cx+y=3$ intersect in the first quadrant.

Q. The equation of the plane passing through the points (2, –4, 0), (3, –4, 5) and parallel to the line 2x = 3y = 4z is :

1. 29x – 27y – 22z = 85
2. 73x + 61y – 22z = 85
3. 125x – 90y – 79z = 340
4. 32x – 21y – 36z = 85

Q.

The equation of the plane which bisects the line joining the points (-1, 2, 3) and (3, -5, 6) at right angle, is

1. 
   

2. 
   

3. 
   

4. 
   

Q. Direction cosines of Z - axis are -

1. 0, 0, 1
2. 0, 1, 0
3. 1, 0, 0
4. 0, 1, 1

Q. The equation of the plane passing through the point $(1,1,1)$ and perpendicular to the planes $2x+y-2z=5\text{and}3x-6y-2z=7$ is

1. $14x+2y-15z=1$
2. $14x-2y+15z=27$
3. $14x+2y+15z=31$
4. $-14x+2y+15z=3$

Q. Equation of the line passing through the point $(2,3,1)$ and parallel to the line of the intersection of the plane $x-2y-z+5=0\text{and}x+y+3z=6$ is

1. $\frac{x-2}{5}=\frac{y-3}{-4}=\frac{z-1}{3}$
2. $\frac{x-2}{-5}=\frac{y-3}{-4}=\frac{z-1}{3}$
3. $\frac{x-2}{5}=\frac{y-3}{4}=\frac{z-1}{3}$
4. $\frac{x-2}{4}=\frac{y-3}{3}=\frac{z-1}{2}$

Q. Find equation of plane passing through (1, 1, 1) and containing the line r= (-3i+j+5k) +$(-3i-j-5k).

Q. The equation of the plane passing through the points (3, 2, -1), (3, 4, 2) and (7, 0, 6) is 5x + 3y – 2z =k , then K is

1. 23
2. 27
3. 21
4. 19

Q. The equation of the plane passing through the points (2, –4, 0), (3, –4, 5) and parallel to the line 2x = 3y = 4z is :

1. 125x – 90y – 79z = 340
2. 32x – 21y – 36z = 85
3. 73x + 61y – 22z = 85
4. 29x – 27y – 22z = 85

Q.

Consider the lines
$L_1:\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1},L_2:\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2},$ and the planes $P_1:7x+y+2z=3,P_2:3x+5y-6z=4.$ Let $ax+by+cz=d$ the equation of the plane passing through the point of intersection of lines $L_1$ and $L_2$ and perpendicular to planes $P_1$ and $P_2$.
Match List I with List II and select the correct answer using the code given below the lists.

$$\begin{array}{cccc}ListI& & ListII& \\ P& a=& 1& 13\\ Q& b=& 2& -3\\ R& c=& 3& 1\\ S& d=& 4& -2\end{array}$$

1. P-3, Q-2, R-4, S-1

2. P-2, Q-4, R-1, S-3

3. P-1, Q-3, R-4, S-2

4. P-3, Q-2, R-1, S-4