InterviewSolution
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InterviewSolution
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The equation of a plane passing through the line of intersection of the planes x+2y+3z = 2 and x y + z = 3 and at a distance 2 3 from the point (3, 1, 1) is (A) 5x 11y + z = 17 (B) 2x y 3 2 1 (C) x + y + z = 3 (D) x 2y 1 2A. `5x-11y+z=17`B. `sqrt(2)x+y=3sqrt(2)-1`C. `x+y+z=sqrt(3)`D. `x-sqrt(y)=1-sqrt(2)` |
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Answer» Correct Answer - A Key Idea (i) Equation on plane through intersection of two planes, i.e. `(a_(1)x+b_(1)y+c_(1)z+d_(1))+lambda` `(a_(2)x+b_(2)y+c _(2)z+d_(2))=0` (ii) Distance of a point `(x_(1),y_(1),z_(1))` from `ax+by+cz+d=0` `=(|ax_(1)+by_(1)+cz_(1)+d|)/(sqrt(a^(2)+b^(2)+c^(2)))` Equation of plane passing through intersection of two planes `x+2y+3z=2" and "x-y+z=3` is `(x+2y+3z-2)+lambda(x-y+z-3)=0` `implies(1+lambda)x+(2-lambda)y+(3+lambda)z-(2+3lambda)=0` whose distance from (3, 1, -1) is `(2)/(sqrt(3))` `implies(|3(1+lambda)+1.(2-lambda)-1(3+lambda)-(2+3lambda)|)/(sqrt((1+lambda)^(2)+(2-lambda)^(2)+(3+lambda)^(2)))=(2)/(sqrt(3))` `implies(|-2lambda|)/(sqrt(3lambda^(2)+4lambda+14))=(2)/(sqrt(2))` `implies" "3lambda^(2)=3lambda^(2)+4lambda+14` `implies" "lambda=-(7)/(2)` `:.(1-(7)/(2))x+(2+(7)/(2))y+(3-(7)/(2))z-(2-(21)/(2))=0` `implies" "-(5x)/(2)+(11)/(2)y-(1)/(2)z+(17)/(2)=0` or`" "5x-11y+z-17=0` |
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