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STATEMENT-1: If three points (x(1),y(1))...

STATEMENT-1: If three points `(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))` are collinear, then `|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0`
STATEMENT-2: If `|{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=0` then the points `(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))` will be collinear.
STATEMENT-3: If lines `a_(1)x+b_(1)y+c_(1)=0,a_(2)=0and a_(3)x+b_(3)y+c_(3)=0` are concurrent then `|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0`

A

T F T

B

T T T

C

F F F

D

F F T .

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Explore conceptually related problems

If |{:(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1):}|=|{:(a_(1),b_(1),1),(a_(2),b_(2),1),(a_(3),b_(3),1):}| , then the two triangles with vertices (x_(1),y_(1)) , (x_(2),y_(2)) , (x_(3),y_(3)) and (a_(1),b_(1)) , (a_(2),b_(2)) , (a_(3),b_(3)) must be congruent.

If the points (x_(1),y_(1)),(x_(2),y_(2)), and (x_(3),y_(3)) are collinear show that (y_(2)-y_(3))/(x_(2)x_(3))+(y_(3)-y_(1))/(x_(3)x_(1))+(y_(1)-y_(2))/(x_(1)x_(2))=0

Knowledge Check

  • Cosnsider the system of equation a_(1)x+b_(1)y+c_(1)z=0, a_(2)x+b_(2)y+c_(2)z=0, a_(3)x+b_(3)y+c_(3)z=0 if |{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=0 , then the system has

    A
    More than two solutions
    B
    One trivial and one-non trivial solutions
    C
    No solution
    D
    Only trivial solution (0,0,0)

  • The lines a_(1)x + b_(1)y + c_(1) = 0 and a_(2)x + b_(2)y + c_(2) = 0 are perpendicular to each other, if

    A
    `a_(1)b_(2) - b_(1)a_(2) = 0`
    B
    `a_(1)a_(2) + b_(1)b_(2) = 0`
    C
    `a_(1)^(2) b_(2) + b_(1)^(2) a_(2) = 0`
    D
    `a_(1)b_(1) + a_(2)b_(2) = 0`

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    If the lines a_(1)x+b_(1)y+1=0,a_(2)x+b_(2)y+1=0 and a_(3)x+b_(3)y+1=0 are concurren ow that the points (a_(1),b_(1)),(a_(2),b_(2)) and (a_(3),b_(3)) are collinear

    If the lines a_(1)x+b_(1)y+1=0,a_(2)x+b_(2)y+1=0 and a_(3)x+b_(3)y+1=0 are concurrent,show that the points (a_(1),b_(1)),(a_(2),b_(2)) and (a_(3),b_(3)) are collinear.

    If the lines a_(1)x+b_(1)y+1=0,a_(2)x+b_(2)y+1=0 and a_(3)x+b_(3)y+1=0 are concurrent,show that the point (a_(1),b_(1)),(a_(1),b_(2)) and (a_(3),b_(3)) are collinear.

    If three points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) lie on the same line,prove that (y_(1)-y_(3))/(x_(2)x_(3))+(y_(3)-y_(1))/(x_(3)x_(1))+(y_(1)-y_(2))/(x_(1)x_(2))=0

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    A(x_(1),y_(1)) , B(x_(2),y_(2)) , C(x_(3),y_(3)) are the vertices of a triangle then the equation |[x,y,1],[x_(1),y_(1),1],[x_(2),y_(2),1]| + |[x,y,1],[x_(1),y_(1),1],[x_(3),y_(3),1]| =0 represents

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