Question

Let Q be the mirror image of the point P(1, 2, 1) with respect to the plane x + 2y + 2z = 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

The Correct Option is B

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)