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 ?

$\text{Given: } Q = 10\hat{i} + 15\hat{j} + 20\hat{k}$

$\text{Plane: } \vec{r} = -(t+p)\hat{i} + t\hat{j} + (1+p)\hat{k}$

$S = \text{reflection of } Q \text{ in the plane}$

$\text{To find: which options are TRUE for } S = \alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$

$x = -(t+p),\quad y = t,\quad z = 1+p$

$t = y,\ p = z-1 \Rightarrow x = -y - (z-1)$

$\Rightarrow\ x + y + z = 1$

$\vec{n} = (1, 1, 1),\qquad |\vec{n}|^2 = 3$

$\text{Plan: reflect } Q \text{ across } x+y+z=1$

$\lambda = \dfrac{Q\cdot\vec{n}-d}{|\vec{n}|^2},\qquad S = Q - 2\lambda\vec{n}$

$Q\cdot\vec{n} = 10+15+20 = 45,\quad d = 1$

$\lambda = \dfrac{45-1}{3} = \dfrac{44}{3}$

$F = Q - \lambda\vec{n} = (10,15,20) - \dfrac{44}{3}(1,1,1)$

$F = \left(-\dfrac{14}{3},\ \dfrac{1}{3},\ \dfrac{16}{3}\right)$

$S = Q - 2\lambda\vec{n} = (10,15,20) - \dfrac{88}{3}(1,1,1)$

$S = \left(-\dfrac{58}{3},\ -\dfrac{43}{3},\ -\dfrac{28}{3}\right)$

$\alpha = -\dfrac{58}{3},\quad \beta = -\dfrac{43}{3},\quad \gamma = -\dfrac{28}{3}$

$3(\alpha+\beta) = -101,\quad 3(\beta+\gamma) = -71$

$3(\gamma+\alpha) = -86,\quad 3(\alpha+\beta+\gamma) = -129$

$\boxed{\text{Answer: (1), (2), (3)}}$