Question

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$\vec{r}=-(t+p)\hat{i}+t\hat{j}+(1+p)\hat{k}$
where $t,p$ are real parameters and $\hat{i},\hat{j},\hat{k}$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat{i}+15\hat{j}+20\hat{k}$ and $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ respectively, then which of the following is/are TRUE ?

• A. $3(\alpha +\beta )=-101$
• B. $3(\beta +\gamma )=-71$
• C. $3(\gamma +\alpha )=-86$
• D. $3(\alpha +\beta +\gamma )=-121$

Answer 1

Answer: (A), (B), (C)

$\overrightarrow{r}=\hat{k}+t(-\hat{i}+\hat{j})+p(-\hat{i}+\hat{k})$
$\overrightarrow{n}=\hat{i}+\hat{j}+\hat{k}$
$\Rightarrow x+y+z=1$
$Q(10,15,20)\text{ and }S(\alpha ,\beta ,\gamma )$
$\frac{\alpha -10}{1}=\frac{\beta -15}{1}=\frac{\gamma -20}{1}=-2\left(\frac{10+15+20-1}{1+1+1}\right)$
$=-\frac{88}{3}$
$\Rightarrow (\alpha ,\beta ,\gamma )\equiv \left(-\frac{58}{3},-\frac{43}{3},-\frac{28}{3}\right)$
$\Rightarrow A,B,C$ are correct options