Image of a Point with Respect to a Plane

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. 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. Let $P$ be a plane passing through the points $(2,1,0),(4,1,1)$ and $(5,0,1)$ and $R$ be any point $(2,1,6)$. Then the image of $R$ in the plane $P$ is:

1. $(6,5,2)$
2. $(6,5,-2)$
3. $(4,3,2)$
4. $(3,4,-2)$

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 $x+y-2z=3$.

The coordinates of $A^{\prime }$ are

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

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

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

1. $(-1,-1,6)$
2. $(2,-1,\frac{13}{2})$
3. $(-2,-1,5)$
4. $(-1,1,10)$

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.

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 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. 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. $(5,10,6)$
2. $(10,15,11)$
3. $(-10,-15,-14)$
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 $x+y-2z=3$.

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