InterviewSolution
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InterviewSolution
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Let `P_(1):2x+y-z=3" and "P_(2):x+2y+z=2` be two planes. Then, which of the following statement(s) is (are) TRUE?A. The line of intersection of `P_(1)` and `P_(2)` has direction ratios 1,2,-1B. The line `(3x-4)/(9)=(1-3y)/(9)=(z)/(3)` is perpendicular to the line of intersection of `P_(1)` and `P_(2)`C. The acute angle between `P_(1)` and `P_(2)` is `60^(@)`D. If `P_(3)` is the plane passing through the point (4,2,-2) and perpendicular to the line of intersection of `P_(1)` and `P_(2)`, then the distance of the point (2,1,1) from the plane `P_(3)" is "(2)/(sqrt(3))` |
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Answer» Correct Answer - C::D We have, `P_(1):2x+y-z=3` and `" "P_(2):x+2y+z=2` Here, `" "vecn_(1)=2hati+hatj-hatk` and`" "vecn_(2)=hati+2hatj+hatk` (a) Direction ratio of the line of intersection of `P_(1)` and `P_(2)` is `thetavecn_(1)xxvecn_(2)` `i.e.|{:(hati,hatj,hatk),(2,1,-1),(1,2,1):}|=(1+2)hati-(2+1)hatj+(4-1)hatk` `=3(hati-hatj+hatk)` Hence, statement a is false. (b) We have, `(3x-4)/(9)=(1-3y)/(9)=(z)/(3)` `implies" "(x-(4)/(3))/(3)=((y-(1)/(3)))/(-3)=(z)/(3)` This line is parallel to the line of intersection of `P_(1)` and `P_(2)`. Hence, statement (b) is false. (c) Let acute angle between `P_(1)` and `P_(2)` be `theta`. We know that, `costheta=(vecn_(1).vecn_(2))/(|vecn_(1)||vecn_(2)|)=((2hati+hatj-hatk).(hati+2hatj+hatk))/(|2hati+hatj-hatk||hati+2hatj+hatk|)` `(2+2-1)/(sqrt(6)xxsqrt(6))=(1)/(2)` `theta=60^(@)` Hence, statement (c) is true. (d) Equation of plane passing through the point (4,2,-2) and perpendicular to the line of intersection of `P_(1)` and `P_(2)` is `3(x-4)-3(y-2)+3(z+2)=0` `implies" "3x-3y+3z-12+6+6=0` `implies" "x-y+z=0` Now, distance of the point (2, 1, 1) from the plane `x-y+z=0` is `D=|(2-1+1)/(sqrt(1+1+1))|=(2)/(sqrt(3))` Hence, statement (d) is true. |
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