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Tardigrade
Question
Mathematics
Let α, β, γ, δ be real numbers such that α2+β2+γ2 ≠ 0 and α+γ=1. Suppose the point (3,2,-1) is the mirror image of the point (1,0,-1) with respect to the plane α x+β y+γ z=δ. Then which of the following statements is/are TRUE?
Q. Let
α
,
β
,
γ
,
δ
be real numbers such that
α
2
+
β
2
+
γ
2
=
0
and
α
+
γ
=
1
. Suppose the point
(
3
,
2
,
−
1
)
is the mirror image of the point
(
1
,
0
,
−
1
)
with respect to the plane
αx
+
β
y
+
γ
z
=
δ
. Then which of the following statements is/are TRUE?
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A
α
+
β
=
2
100%
B
δ
−
γ
=
3
200%
C
δ
+
β
=
4
300%
D
α
+
β
+
γ
=
δ
0%
Solution:
Direction ratio of line joining
(
3
,
2
,
−
1
)
&
(
1
,
0
,
−
1
)
is
(
2
,
2
,
0
)
proportional to
(
α
,
β
,
γ
)
⇒
γ
=
0
,
α
+
γ
=
1
,
α
=
1
,
β
=
1
So plane
x
+
y
=
δ
(
2
,
1
,
−
1
)
satisfies if
δ
=
3