If four points `(x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))` and `(x_(4),y_(4))` taken in order in a parallelogram, then:

A(6,1), B(8,2) and C(9,4) are three vertices of a parallelograam ABCd taken in order. Find the fourth vertex D. IF (x_(1),y_(1)), (x_(2),y_(2)), (x_(3),y_(3)) and (x_(4),y_(4)) are the four vertices of the parallelgraam then using the given points, find the value of (x_(1)+x_(3)-x_(2),y_(1)+y_(3)+y_(2)) and state the reason for your result.

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)) and S(x_(4),y_(4)) then

If (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))and(x_(4),y_(4)) be the consecutive vertices of a parallelogram , show that , x_(1)+x_(3)=x_(2)+x_(4)andy_(1)+y_(3)=y_(2)+y_(4) .

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

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 which of the following need not hold. (a) x_(1)+x_(2)+x_(3)+x_(4)=0 (b) x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=c^(4) (c) y_(1)+y_(2)+y_(3)+y_(4)=0 (d) x_(1)+y_(2)+x_(3)+y_(4)=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 which of the following need not hold. (a) x_(1)+x_(2)+x_(3)+x_(4)=0 (b) x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=c^(4) (c) y_(1)+y_(2)+y_(3)+y_(4)=0 (d) x_(1)+y_(2)+x_(3)+y_(4)=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 which of the following need not hold. (a) x_(1)+x_(2)+x_(3)+x_(4)=0 (b) x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=c^(4) (c) y_(1)+y_(2)+y_(3)+y_(4)=0 (d) x_(1)+y_(2)+x_(3)+y_(4)=0

If the points of intersection of the parabola y^(2) = 4ax and the circle x^(2) + y^(2) + 2gx + 2fy + c = 0 are (x_(1), y_(1)) ,(x_(2) ,y_(2)) ,(x_(3), y_(3)) and (x_(4), y_(4)) respectively , then _