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Mathematics Question on Three Dimensional Geometry

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

A

3(α+β)=−1013(\alpha+\beta)=-101

B

3(β+γ)=−713(\beta+\gamma)=-71

C

3(γ+α)=−863(\gamma+\alpha)=-86

D

3(α+β+γ)=−1213(\alpha+\beta+\gamma)=-121

Answer

3(γ+α)=−863(\gamma+\alpha)=-86

Explanation

Solution

Given :
Equation of the plane :
r⃗=−(t+p)i^+tj^+(1+p)k^\vec{r}=-(t+p) \hat{ i }+t \hat{ j }+(1+p) \hat{ k }
r⃗=k^+t(−i^+j^)+p(−i^+k^)\vec{r}=\hat{k}+t(-\hat{i}+\hat{j})+p(-\hat{i}+\hat{k})
Standard form of Equation of plane :
[r⃗−k^          i^+j^         −i^+k^]=0[\vec{r}-\hat{k}\ \ \ \ \ \ \ \ \ \ \hat{i}+\hat{j}\ \ \ \ \ \ \ \ \ -\hat{i}+\hat{k}]=0
Therefore, x + y + z = 1 ……. (i)
Coordinates of Q and S :
Q = (10, 15, 20)
S = (α, β, γ)
∴ ⇒α−101=β−151=γ−201⇒\frac{α-10}{1}=\frac{β-15}{1}=\frac{γ-20}{1}
=−2(10+15+20−1)3=\frac{-2(10+15+20-1)}{3}
∴ α = 10 = β = -15 γ - 20 = −833-\frac{83}{3}
Therefore, the values are as follows :
α=−583, β=−433,γ=−833α=-\frac{58}{3},\ β=-\frac{43}{3},γ=-\frac{83}{3}
∴ 3 (α + β) = −101 so, option (A) is correct.
3(β + γ) =−71 so, option (B) is correct.
3(γ + α) = −86 so, option (C) is correct.
3(α+β+γ)=−129 so, option (D) is incorrect.

So, the correct options are (A), (B) and (C).