If the circle `x^(2)+y^(2)=a^(2)` intersects the hyperbola `xy=c^(2)` at 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 the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then x_1+x_2+x_3+x_4=0 y_1+y_2+y_3+y_4=0 x_1x_2x_3x_4=C^4 y_1y_2y_3y_4=C^4

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.

The circle x^(2) + y^(2) = 4 x + 8 y + 5 intersects the line 3x -4y =m at two distinct points if

If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the vlaue of (1)/(y_(1))+(1)/(y_(2)) is

The line y=mx+1 touches the curves y=0x^(4)+2x^(2)+x at two points P(x_(1),y_(1)) and Q(x_(2),y_(2)) . The value of x_(1)^(2)+x_(2)^(2)+y_(1)^(2)+y_(2)^(2) is

If x_(1),x_(2),x(3) as well as y_(1),y_(2),y_(3) are in geometric progression with the same common ratio,then the points (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3))

If x_(1), x_(2), x_(3) and y_(1),y_(2), y_(3) are in arithmetic progression with the same common difference then the points (x_(1),y_(1)) (x_(2),y_(2)) (x_(3), y_(3)) are:

If three point A(x_(1), y_(1)), B(x_(2), y_(2)) and C(x_(3), y_(3)) will be collinear then the area of triangleABC= ____.

if (x_(1),x_(2))^(2)+(y_(1)-y_(2))^(2)=a^(2), (x_(2)-x_(3))^(2)+(y_(2)-y_(3))^(2)=b^(2) (x_(3)-x_(1))^(2)+(y_(3)-y_(1))^(2)=c^(2). where a,b,c are positive then prove that 4 |{:(x_(1),,y_(1),,1),(x_(2) ,,y_(2),,1),( x_(3),, y_(3),,1):}| = (a+b+c) (b+c-a) (c+a-b)(a+b-c)