Question 275 of 521MediumMCQ
The vector equation of the plane passing through the intersection of the planes r⋅(i^+j^+k^)=1 and r⋅(i^−2j^)=−2, and the point (1,0,2), is:
Solutions · 1
Official solution
Given, point (1, 0, 2)
Equation of plane =
\[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) = 1\] and \[\overrightarrow r\,.\,(\widehat i - 2\widehat j) = - 2\]
Equation of plane passing through the intersection of given planes is
\[[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \lambda [\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0\]
\[\because\] This plane passes through point (1, 0, 2) i.e.,
vector \[(\widehat i + 2\widehat k)\]
\[\therefore\] \[[(\widehat i + 2\widehat k)\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \lambda [(\widehat i + 2\widehat k)\,.\,(\widehat i - 2\widehat j) + 2] = 0\]
\[ \Rightarrow (3 - 1) + \lambda (1 + 2) = 0\]
\[ \Rightarrow 2 + \lambda \times 3 = 0\]
\[ \Rightarrow \lambda = - 2/3\]
Hence, equation of required plane is
\[[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] + \left( {{{ - 2} \over 3}} \right)[\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0\]
\[ \Rightarrow \] \[3[\overrightarrow r\,.\,(\widehat i + \widehat j + \widehat k) - 1] - 2[\overrightarrow r\,.\,(\widehat i - 2\widehat j) + 2] = 0\]
\[ \Rightarrow \] \[\overrightarrow r\,.\,(\widehat i + 7\widehat j + 3\widehat k) = 7\]
Answer the question to unlock