Question

Let the mirror image of the point $\left(1,3,a\right)$ with respect to the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$ be $\left(-3,5,2\right)$. Then, the value of $\left|a+b\right|$ is equal to ______.

Answer 1

Step-1 Using mid point concept :

Given the points on the plane are $P\left(1,3,a\right)$ and $Q\left(-3,5,2\right)$ also the equation of the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$.

Let $R\left(x,y,z\right)$ be the mid-point of the line $PQ$ which lies on the plane.

Find the coordinate of $R$ using the mid-point formula,

$$\begin{gathered} \Rightarrow x=\frac{1-3}{2}=-\frac{2}{2}=-1 \\ \Rightarrow y=\frac{3+5}{2}=\frac{8}{2}=4 \\ \Rightarrow z=\frac{a+2}{2} \end{gathered}$$

Therefore, the coordinate of $R$ is $\left(-1,4,\frac{a+2}{2}\right)$.

Since the point $R$ lies on the plane so substitute the points of $R$ in the plane equation $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$.

$$\begin{gathered} \Rightarrow 2\left(-1\right)-4+\left(\frac{a+2}{2}\right)-b=0 \\ \Rightarrow -2-4+\left(\frac{a+2}{2}\right)-b=0 \\ \Rightarrow -12+a+2-2b=0 \\ \Rightarrow a=2b+10 \end{gathered}$$

Step-2 Finding the value of $\left|a+b\right|$ :

And the direction ratio of $PQ$ is $\left(4,-2,a-2\right)$.

$\Rightarrow \frac{2}{4}=\frac{-1}{-2}=\frac{1}{a-2}$

Now simplify for the value of $a$ and $b$.

$$\begin{gathered} \Rightarrow a-2=2 \\ \Rightarrow a=4 \end{gathered}$$

Substitute the value of $a$ as $4$ in $a=2b+10$.

$$\begin{gathered} \Rightarrow 4=2b+10 \\ \Rightarrow 4-10=2b \\ \Rightarrow -6=2b \\ \Rightarrow b=-3 \end{gathered}$$

Substitute the values of $a$ and $b$ in $\left|a+b\right|$.

$$\begin{gathered} \Rightarrow \left|a+b\right|=\left|4-3\right| \\ \Rightarrow \left|a+b\right|=1 \end{gathered}$$

Therefore, the value of $\left|a+b\right|$ is equal to $1$.