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NCERT Class XII Mathematics Chapter - - Solutions

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Question : 67 of 78
Marks: +1, -0
Find the equation of the plane passing through (a, b, c) and parallel to the plane r⃗⋅(i^+j^+k^)=2.\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2 .
Solution:  
Any plane parallel to r⃗⋅(i^+j^+k^)=2\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2 is r⃗⋅(i^+j^+k^)=λ\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = \lambda ...(1)
This passes through the point (a, b, c)
i.e., (ai^+bj^+ck^)(\widehat{ai} + \widehat{bj} + \widehat{ck})
∴(ai^+bj^+ck^)⋅(i^+j^+k^)=λ\therefore (\widehat{ai} + \widehat{bj} + \widehat{ck}) \cdot (\hat{i} + \hat{j} + \hat{k}) = \lambda
⇒(a)(1)+(b)(1)+(c)(1)=λ\Rightarrow (a)(1) + (b)(1) + (c)(1) = \lambda
⇒λ=a+b+c\Rightarrow \lambda = a + b + c
Putting the value of l in (1), the required equation of the plane is r⃗⋅(i^+j^+k^)=a+b+c\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = a + b + c
⇒(xi^+yj^+zk^)⋅(i^+j^+k^)\Rightarrow (\widehat{xi} + \widehat{yj} + \widehat{zk}) \cdot (\hat{i} + \hat{j} + \hat{k}) =a+b+c= a + b + c
⇒x+y+z=a+b+c.\Rightarrow x + y + z = a + b + c.
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