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