InterviewSolution
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InterviewSolution
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The driection ratios of normal to the plane through the points (0, -1, 0) and (0, 0, 1) and making an angle `pi//4` with the plane `y-z+5=0` areA. 2, -1, 1B. `sqrt(2), 1, -1`C. `2, sqrt(2), -sqrt(2)`D. `2sqrt(3), 1, -1` |
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Answer» Correct Answer - B::C Let the equation of plane be `a(x-0)+b(y+1)+c(z-0)=0` [`because` Equation of plane passing through a point `(x_(1),y_(1),z_(1))` is given by `a(x-x_(1))+b(y-y_(1))+c(z-z_(1))=0`] `impliesax+by+cz+b=0" "...(i)` Since, it also passes through (0, 0, 1) therefore, we get `c+b=0" "(ii)` Now, as angle between the planes `ax+by+cz+b=c` and `" "y-z+5=0" is "(pi)/(4).` `:.cos((pi)/(4))=(|n_(1).n_(2)|)/(|n_(1)||n_(2)|)`, where `n_(1)=ahati+bhatj+chatk` and `n _(2)=0hati+hatj+hatk` `implies" "(1)/(sqrt(2))=(|(ahati+bhatj+chatk).(0hati+hatj-hatk)|)/(sqrt(a^(2)+b^(2)+c^(2))sqrt(0+1+1))` `=(|b-c|)/(sqrt(a^(2)+b^(2)+c^(2))sqrt(2))` `impliesa^(2)+b^(2)+c^(2)=|b-c|^(2)=(b-c)^(2)=b^(2)+c^(2)-2bc` `implies" "a^(2)=-2bc` `implies" "a^(2)=2b^(2)" "["Using Eq. (ii)"]` `implies" "a=+-sqrt(2)b` `implies"Direction ratios (a, b, c)"=(+-sqrt(2), 1, -1)` So, options (b) and (c) are correct because `2, sqrt(2), -sqrt(2)` and `sqrt(2), 1, -1`. are multiple of each other. |
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