Image of a Point with Respect to a Plane

Q. Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of $PQ$ lies in the plane $x+y=3$) lies on the $z-$axis. Let the distance of $P$ from the $x-$axis be $5.$ If $R$ is the image of $P$ in the $xy-$plane, then the length of $PR$ is

Q.

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane 2x - y + z + 3 = 0 is the line

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

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

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

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

Q.

Let$\alpha ,\beta ,\gamma ,\delta$ be real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2\ne 0$ and $\alpha +\gamma =1$.Suppose the point $(3,2,-1)$ is the mirror image of the point$(1,0,-1)$ with respect to the plane $\alpha x+\beta y+\gamma z=\delta$. Then which of the following statements is/are TRUE?

1. $\alpha +\beta =2$

2. $\delta –\gamma =3$

3. $\delta +\beta =4$

4. $\alpha +\beta +\gamma =\delta$

Q.

If $\left(1,-2,-2\right)$ and $\left(0,2,1\right)$ are direction ratios of two lines, then the direction cosines of a perpendicular to both the lines are

1. $\left(\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right)$

2. $\left(\frac{2}{3},-\frac{1}{3},\frac{2}{3}\right)$

3. $\left(-\frac{2}{3},-\frac{1}{3},\frac{2}{3}\right)$

4. $\left(\frac{2}{\sqrt{14}},-\frac{1}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$

Q.

The equation of the plane passing through the point (3, - 3, 1) and perpendicular to the line joining the points (3, 4, - 1) and (2, - 1, 5) is:

1. x - 5y + 6z + 18 = 0

2. x + 5y - 6z + 18 = 0

3. - x - 5y - 6z + 18 = 0

4. - x - 5y + 6z + 18 = 0

Q.

Find the image of :

(i) (-2, 3, 4) in the yz-plane.

(ii) (-5, 4, -3) in the xz-plane.

(iii) (5, 2, -7) in the xy-plane.

(iv)(-5, 0, 3) in the xz-plane.

(v)(-4, 0, 0) in the xy-plane.

Q. Find the image of the point P (3, 5, 7) in the plane 2x+y+z = 0 ?

1. (-3, 2, 4)
2. (-9, -1, 1)
3. (-1, -9, 1)
4. None of these

Q.

The coordinates of the foot of the perpendicular from $(0,0)$upon the line $x+y=2$ are

1. $(2,-1)$

2. $(-2,1)$

3. $(1,1)$

4. $(1,2)$

Q. If a line passes through two points (1, 2, 3) & (4, 5, 6) then the direction ratios of that line would be -

1. 1, 2, 3
2. 3, 2, 1
3. 3, 4, 5
4. 3, 3, 3

Q. A ray of light comes along the line $L=0$ and strikes the plane mirror kept along the plane $P=0$ at $B$. $A(2,1,6)$ is a point on the line $L=0$ whose image about $P=0$ is $A^{\prime }$. It is given that $L=0$ is $\frac{x-2}{3}=\frac{y-1}{4}=\frac{z-6}{5}$ and $P=0$ is $x+y-2z=3$.

If $L_1=0$ is the reflected ray, then its equation is

1. $\frac{x+10}{4}=\frac{y-5}{4}=\frac{z+2}{3}$
2. $\frac{x+10}{3}=\frac{y+15}{5}=\frac{z+14}{5}$
3. $\frac{x+10}{4}=\frac{y+15}{5}=\frac{z+14}{3}$
4. None of these

Q. A ray of light comes along the line $L=0$ and strikes the plane mirror kept along the plane $P=0$ at $B$. $A(2,1,6)$ is a point on the line $L=0$ whose image about $P=0$ is $A^{\prime }$. It is given that $L=0$ is $\frac{x-2}{3}=\frac{y-1}{4}=\frac{z-6}{5}$ and $P=0$ is $x+y-2z=3$.

The coordinates of $B$ are

1. $(10,15,11)$
2. $(-10,-15,-14)$
3. None of these
4. $(5,10,6)$

Q. The image of the point (-1, 3, 4) in the plane x-2y=0 is

1. $(15,11,14)$
2. $(-\frac{17}{3},-\frac{19}{3},1)$
3. $(\frac{9}{5},-\frac{13}{5},4)$
4. $(-\frac{17}{3},-\frac{19}{3},4)$

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

1. $3\sqrt{5}$
2. $2\sqrt{42}$
3. $\sqrt{42}$
4. $6\sqrt{5}$

Q. The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2x-y+z+3=0$ is the line:

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

Q.

The equation of plane passing through a point $A(2,-1,3)$ and parallel to the vectors $a=(3,0,-1)andb=(-3,2,2)$ is:

Q. Line $L$ lies in the plane $x+3y-z=9$ is perpendicular to line $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (2\hat{i}+\hat{j}-\hat{k}),\lambda \in R$ and passes through the point where plane meets the given line. Sum of coordinates of the point, where the line meets the $XY$ plane is

Q. The reflection of the point $\overrightarrow{a}$ in the plane $\overrightarrow{r}\cdot \overrightarrow{n}=q$ is

1. $\overrightarrow{a}+\frac{(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{|\overrightarrow{n}|}$
2. $\overrightarrow{a}+\frac{2(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{|\overrightarrow{n}{|}^2}\overrightarrow{n}$
3. $\overrightarrow{a}+\frac{2(\overrightarrow{q}-\overrightarrow{a}\cdot \overrightarrow{n})}{|\overrightarrow{n}|}\overrightarrow{n}$
4. none of these

Q. Write direction cosines of a line parallel to z-axis.

Q. The equation of the plane bisecting the angle between the planes 3x+4y=4 and 6x−2y+3z+5=0 that contains the origin, is

1. $51x+18y+15z-3=0$
2. $9x-38y-15z+53=0$
3. $9x-2y-3z+13=0$
4. $51x+8y+5z-3=0$

Q.

Distance of the point (2, 3, 4) from the plane $3x-6y+2z+11=0$ is

1. 2

2. 3

3. 0

4. 1

Q. A plane passes through the points P (1, 1, 1) , Q (3, -1, 2) and R (-3, 5 , -4). The equation of the plane is .

1. $6x+16y+16z-44=0$
2. $x-2y+16z-13=0$
3. $6x+6y+16z-28=0$
4. $x+6y+6z-22=0$

Q. If the image of the point $P(1,-2,3)$ 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. The image of the point $(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$

1. $(-1,0,7)$
2. $(1,0,7)$
3. $(2,1,5)$
4. $(2,1,-5)$

Q.

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane 2x - y + z + 3 = 0 is the line

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

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

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

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

Q. Let $R$ be a rectangle, $C$ be a circle, and $T$ be a triangle in the plane. The maximum possible number of points common to the perimeters of $R,C,$ and $T$ is

1. 4
2. 3
3. 5
4. 6

Q.

If the image of the point P(1, -2, 3) 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. $3\sqrt{5}$

2. $2\sqrt{42}$

3. $\sqrt{42}$

4. $6\sqrt{5}$

Q.

The point of intersection of the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z+2}{3}$ and the plane 2x + 3y + z = 0 is
[MP PET 1989]

1. (0, 1, -2)

2. (1, 2, 3)

3. (-1, 9, -25)

4. 
   

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

1. $(-2,2,8)$
2. $(-3,6,13)$
3. $(2,2,8)$
4. $(3,6,13)$

Q. What are the direction cosines of X-axis?

Q. Find the equation of the perpendicular drawn from the point P (2, 4, −1) to the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}.$ Also, write down the coordinates of the foot of the perpendicular from P.