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CBSE Class 12 Math 2011 Solved Paper

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Question : 27
Total: 29
Find the equation of the plane which contains the line of intersection of the planes
→
r
.(
^
i
+2
^
j
+3
^
k
) - 4 = 0 ,
→
r
.(2
^
i
+
^
j
−
^
k
) + 5 = 0 and which is perpendicular to the plane
→
r
.(5
^
i
+3
^
j
−6
^
k
) + 8 = 0
Solution:  
The equations of the given planes are
→
r
.(
^
i
+2
^
j
+3
^
k
) - 4 = 0 ... (1)
→
r
.(2
^
i
+
^
j
−
^
k
) + 5 = 0 ... (2)
The equation of the plane passing through the line of intersection of the given planes is
[
→
r
.(
^
i
+2
^
j
+3
^
k
)−4] + λ [
→
r
.(2
^
i
+
^
j
−
^
k
)+5] = 0
→
r
 [(1 + 2λ)
^
i
 + (2 + λ)
^
j
 + (3 - λ)
^
k
] + (- 4 + 5λ) = 0 ... (3)
The plane in equation (3) is perpendicular to the plane,
→
r
.(5
^
i
+3
^
j
−6
^
k
) + 8 = 0
∴ 5 (1 + 2λ) + 3 (2 + λ) - 6 (3 - λ) = 0
⇒ 5 + 10λ + 6 + 3λ - 18 + 6λ = 0
⇒ 19λ - 7 = 0
⇒ λ =
7
19
Substituting λ =
7
19
 in equation (3),
→
r
.[
33
19
^
i
+
45
19
^
j
+
50
19
^
k
] -
41
19
 = 0
⇒
→
r
.(33
^
i
+45
^
j
+50
^
k
) - 41 = 0
This is the vector equation of the required plane.
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