Let S be the reflection of a point Q with respect to the plane given by
$\vec{r} = (2 p - t + 1) \hat{i} + \hat{j} (1 + 2 t + p) + (1 - t - 3 p) \hat{k}$
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 $10 \hat{i} + 15 \hat{j} + 19 \hat{k}$ and $α \hat{i} + β \hat{j} + γ \hat{k}$ respectively, then which of the following is/are TRUE?

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$3 (α + β + γ) = - 121$

$3 (α + β) = - 89$

$3 (β + γ) = - 62$

$3 (γ + α) = - 77$

answer is A, B, C.

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Detailed Solution

$\vec{r} = \hat{k} + t (- \hat{i} + \hat{j}) + p (- \hat{i} + \hat{k})$
$\vec{n} = \hat{i} + \hat{j} + \hat{k}$
$\Rightarrow x + y + z = 1$
$Q (10, 15, 20)$ , and $S (α, β, γ)$
$\frac{α - 10}{1} = \frac{β - 15}{1} = \frac{γ - 20}{1} = 2 (\frac{10 + 15 + 20 - 1}{1 + 1 + 1})$
$= - \frac{88}{3}$
$\Rightarrow (α, β, γ) \equiv (- \frac{58}{3}, - \frac{43}{3}, - \frac{28}{3})$
$\Rightarrow$ A, B, C are correct options