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JEE Main Math Class 12 Three Dimensional Geometry Part 2 Questions
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© examsnet.com
Question : 60
Total: 100
The vector equation of the plane passing through the intersection of the planes
r
⋅
(
∧
i
+
j
+
∧
k
)
=
1
and
r
⋅
(
∧
i
−
2
∧
j
)
=
−
2
and the point
(
1
,
0
,
2
)
is
[24 Feb 2021 Shift 2]
r
⋅
(
∧
i
+
7
∧
j
+
3
∧
k
)
=
7
3
r
⋅
(
3
∧
i
+
7
∧
j
+
3
∧
k
)
=
7
r
⋅
(
∧
i
+
7
∧
j
+
3
∧
k
)
=
7
r
⋅
(
∧
i
−
7
∧
j
+
3
∧
k
)
=
7
3
Validate
Solution:
Given, point
(
1
,
0
,
2
)
Equation of plane
=
r
⋅
(
∧
i
+
∧
j
+
∧
k
)
=
1
and
r
⋅
(
∧
i
−
2
∧
j
)
=
−
2
Equation of plane passing through the intersection of given planes is
[
r
⋅
(
∧
i
+
∧
j
+
∧
k
)
−
1
]
+
λ
[
r
⋅
(
∧
i
−
2
∧
j
)
+
2
]
=
0
∵
This plane passes through point
(
1
,
0
,
2
)
i.e.,
v
e
c
t
o
r
(
∧
i
+
2
∧
k
)
∴
[
(
∧
i
+
2
∧
k
)
⋅
(
∧
i
+
∧
j
+
∧
k
)
−
1
]
+
λ
[
(
∧
i
+
2
∧
k
)
⋅
(
∧
i
−
2
∧
j
)
+
2
]
=
0
⇒
(
3
−
1
)
+
λ
(
1
+
2
)
=
0
⇒
2
+
λ
×
3
=
0
⇒
λ
=
−
2
∕
3
Hence, equation of required plane is
[
r
⋅
(
∧
i
+
∧
j
+
∧
k
)
−
1
]
+
(
−
2
3
)
[
r
⋅
(
∧
i
−
2
∧
j
)
+
2
]
=
0
or
3
[
r
⋅
(
∧
i
+
∧
j
+
∧
k
)
−
1
]
−
2
[
r
⋅
(
∧
i
−
2
∧
j
)
+
2
]
=
0
or
r
⋅
(
∧
i
+
7
∧
j
+
3
∧
k
)
=
7
© examsnet.com
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