[" The points "],[(x_(1),y_(1)),(x_(2),y_(2)),(x_(1),y_(2))" and "],[(x_(2),y_(1))" are always: "]

Four points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)) and (x_(4),y_(4)) are such that sum_(i=1)^(4)(x_(i)^(2)+y_(i)^(2))<=2(x_(1)x_(3)+x_(2)x_(4)+y_(1)y_(2)+y_(3)y_(4)) Then these points are vertices of - (A) Parallelogram (B) Rectangle (C) Square (D) Rhombus

If _(i-1)(x_(i)^(2)+y_(i)^(2))<=2x_(1)x_(3)+2x_(2)x_(4)+2y_(2)y_(3)+2y_(1)y_(4)sum_(i-1)^(4)(x_(i)^(2)+y_(i)^(2))<=2x_(1)x_(3)+2x_(2)x_(4)+2y_(2)y_(3)+2y_(1)y_(4) the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),(x_(4),y_(4)) are the vertices of a rectangle collinear the vertices of a trapezium none of these

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

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

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

If the circle x^(2)+y^(2)=a^(2) intersects the hyperbola xy=c^(2) in four points P (x_(1) ,y_(1)) Q (x_(2), y_(2)) R (x_(3) ,y_(3)) S (x_(4) ,y_(4)) then 1) x_(1)+x_(2)+x_(3)+x_(4)=2c^(2) 2) y_(1)+y_(2)+y_(3)+y_(4)=0 3) x_(1)x_(2)x_(3)x_(4)=2c^(4) 4) y_(1)y_(2)y_(3)y_(4)=2c^(4)

Prove that the line passing through the points (x_(1),y_(1)) and (x_(2),y_(2)) is at a distance of |(x_(1)y_(2)-x_(2)y_(1))/(sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)))| from origin.

Prove that the line passing through the points (x_(1),y_(1)) and (x_(2),y_(2)) is at a distance of |(x_(1)y_(2)-x_(2)y_(1))/(sqrt((x_(2)-x_(1))^(2)+(y_(2)-y_(1))^(2)))| from origin.