Question

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2 y + 2 z = 16. Let T be a plane passing through the point Q and contains the line
$\vec{r}=−\hat{k}+λ(\hat{i}+\hat{j}+2\hat{k}), λ ∈ R.$
Then, which of the following points lies on T?

• A. (2, 1, 0)
• B. (1, 2, 1)
• C. (1, 2, 2)
• D. (1, 3, 2)

Answer 1

Answer: (B)

(1, 2, 1)

Answer 2

The correct answer is (B) : (1, 2, 1)
P(1,2,1) image in plane x+2y+2z = 16
$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-1}{2} = -\frac{2(1+2\times2+2\times1-16)}{1^2+2^2+2^2}$
$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-1}{2} = 2$
Q(3,6,5)
$\vec{r}=−\hat{k}+λ(\hat{i}+\hat{j}+2\hat{k})$

Fig.

$AQ = 3\hat{i}+ 6\hat{j} + 6\hat{k}$
$=3(\hat{i}+2\hat{j}+2\hat{k})$
$\vec{n}= 3(\hat{i}+ 2\hat{j} + 2\hat{k}) × (\hat{i} + \hat{j} + 2\hat{k})$
$\begin{vmatrix} \hat{i} & \hat{j}& \hat{k} \\\ 1 & 2 & 2 \\\ 1 & 1 & 2 \\\ \end{vmatrix}$
$= 2\hat{i} - 0\hat{j} - \hat{k}$
Equation of plane = 2(x-0) + 0(y-0) -1(z+1) = 0
2x-z=1
Point lying on plane is (1,2,1)