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NCERT Class XII Mathematics Chapter - - Solutions

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Question : 74 of 78
Marks: +1, -0
Find the equation of the plane passing through the line of intersection of the planes r→⋅(i^+j^+k^)=1\overset{\rightarrow}{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1 and r→⋅(2i^+3j^−k^)+4=0\overset{\rightarrow}{r} \cdot (\widehat{2i} + \widehat{3j} - \hat{k}) + 4 = 0 and parallel to x-axis.
Solution:  
The given planes are x + y + z – 1 = 0 ...(1)
and 2x + 3y – z + 4 = 0 ...(2)
Any plane through their intersection is
(x + y + z – 1) + k ( 2x + 3y – z + 4) = 0 ...(3)
Since it is parallel to x-axis,
∴ Normal to (3) is perpendicular to x-axis
⇒ (1 + 2k) (1) + (1 + 3k) (0) + (1 – k) (0) = 0
⇒1+2k=0⇒k=−12\Rightarrow 1 + 2k = 0 \Rightarrow k = -\frac{1}{2}
Putting in (3), we get (x+y+z−1)(x + y + z - 1) −12(2x+3y−z+4)=0-\frac{1}{2}(2x + 3y - z + 4) = 0
⇒ 2x + 2y + 2z – 2 – 2x – 3y + z – 4 = 0
⇒ –y + 3z – 6 = 0
⇒ y – 3z + 6 = 0.
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