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Byju's Answer
Standard XII
Mathematics
Condition for Two Lines to be on the Same Plane
Prove that th...
Question
Prove that the lines
x
−
2
1
=
y
−
4
4
=
z
−
6
7
and
x
+
1
3
=
y
+
3
5
=
z
+
5
7
are coplanar. Also, find the equation of the plane containing these two lines.
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Solution
If two line are coplanar then
∣
∣ ∣
∣
x
2
−
x
1
y
2
−
y
1
z
2
−
z
1
1
1
m
1
n
1
l
2
m
2
n
2
∣
∣ ∣
∣
=
0
Here
(
x
1
,
y
1
,
z
1
)
=
(
2
,
4
,
6
)
(
x
2
,
y
2
,
z
2
)
=
(
−
1
,
−
3
,
−
5
)
(
l
1
,
m
1
,
n
1
)
=
(
1
,
4
,
7
)
(
l
2
,
m
2
,
n
2
)
=
(
3
,
5
,
7
)
Substituting the values
=
∣
∣ ∣
∣
−
3
−
7
−
11
1
4
7
3
5
7
∣
∣ ∣
∣
=
−
3
(
28
−
35
)
+
7
(
7
−
21
)
−
11
(
5
−
12
)
=
21
−
98
+
77
=
98
−
98
=
0
Hence the lines are coplanar.
The equation of the plane containing these lines is
∣
∣ ∣
∣
x
−
2
y
−
4
z
−
6
1
4
7
3
5
7
∣
∣ ∣
∣
=
0
⇒
(
x
−
2
)
(
28
−
35
)
−
(
y
−
4
)
(
7
−
21
)
+
(
z
−
6
)
(
5
−
12
)
=
0
⇒
(
x
−
2
)
(
−
7
)
−
(
y
−
4
)
(
−
14
)
+
(
z
−
6
)
(
−
7
)
=
0
⇒
−
7
x
+
14
+
14
y
−
56
−
7
z
+
42
=
0
⇒
−
7
x
+
14
y
−
7
z
=
0
⇒
x
−
2
y
+
z
=
0
Hence this is the equation of the plane.
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Similar questions
Q.
Show that the lines
x
+
3
−
3
=
y
−
1
1
=
z
−
5
5
;
x
+
1
−
1
=
y
−
2
2
=
z
−
5
5
are coplanar. Also find the equation of plane containing the lines.
Q.
Show that the lines
x
+
1
-
3
=
y
-
3
2
=
z
+
2
1
and
x
1
=
y
-
7
-
3
=
z
+
7
2
are coplanar. Also, find the equation of the plane containing them.
Q.
show that the lines
x
+
3
−
3
=
y
−
1
1
=
z
−
5
5
and
x
+
1
−
1
=
y
−
2
2
=
z
−
5
5
are coplanar. Also find the equation of the plane.
Q.
Find the equation of a plane containing the lines
x
−
5
4
=
y
−
7
4
=
z
+
3
−
5
and
x
−
8
7
=
y
−
4
1
=
z
−
5
3
.
Q.
Show that the lines
x
−
4
5
=
y
−
3
−
2
=
z
−
2
−
6
and
x
−
3
4
=
y
−
2
−
3
=
z
−
1
−
7
are coplanar. Find their point of intersection and the equation of the plane in which they lie.
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