Image of a Point with Respect to a Plane

Q. A tetrahedron $\left(T_1\right)$ has a fixed volume, $V_1$. Another tetrahedron $\left(T_2\right)$ with volume $V_2$ is drawn by joining the centroids of the faces of $T_1.$ If $V_1=k{V}_2,$ then the value of $k$ is

1. $\frac{9}{2}$
2. $18$
3. $27$
4. $9$

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

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

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. Image point of (1, 3, 4) in the plane 2x - y + z + 3 = 0 is

1. (-3, 5, 2)
2. (3, 5, -2)
3. None of these
4. (3, -5, 3)

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.

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.

Find the equation of the plane for which the mirror of (2, 2, 2) is (-2, 0, -2)

1. 2x+8y+2z-8=0
2. -2x-2z=0
3. 2x+3y+2z=3
4. 2x+y+2z-1=0

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. 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 $A^{\prime }$ are

1. $(6,-5,2)$
2. None of these
3. $(6,5,2)$
4. $(6,5,-2)$

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