If a circle and the rectangular hyperbola `xy = c^(2)` meet in four points `'t'_(1) , 't'_(2) , 't'_(3) " and " 't'_(4)` then prove that `t_(1) t_(2) t_(3) t_(4) = 1`.

Statement :1 If a parabola y ^(2) = 4ax intersects a circle in three co-normal points then the circle also passes through the vertr of the parabola. Because Statement : 2 If the parabola intersects circle in four points t _(1), t_(2), t_(3) and t_(4) then t _(1) + t_(2) + t_(3) +t_(4) =0 and for co-normal points t _(1), t_(2) , t_(3) we have t_(1)+t_(2) +t_(3)=0.

If a circle cuts a rectangular hyperbola xy=1 in four points P,Q,R,S and the parameters of these four points be t_(1),t_(2),t_(3) and t_(4) respectively and -20t_(1)t_(2)t_(3)t_(4)=k , then value of k equals

The normal to the rectangular hyperbola xy=4 at the point t_(1), meets the curve again at the point t_(2) Then the value of t_(1)^(2)t_(2) is

If the normal to the rectangular hyperbola xy=c^(2) at the point t' meets the curve again at t_(1) then t^(3)t_(1), has the value equal to

Let a circle x^(2) + y^(2) + 2gx + 2fy + k = 0 cuts a rectangular hyperbola xy = c^(2) in (ct_(1), c/t_(1)) , (ct_(2), c/t_(2)) , (ct_(3), c/t_(3)) " and " (ct_(4), c/t_(4)) .

If the normal to the rectangular hyperbola xy = 4 at the point (2t, (2)/(t_(1))) meets the curve again at (2t_(2), (2)/(t_(2))) , then

If a circle intersects the parabola y^(2) = 4ax at points A(at_(1)^(2), 2at_(1)), B(at_(2)^(2), 2at_(2)), C(at_(3)^(2), 2at_(3)), D(at_(4)^(2), 2at_(4)), then t_(1) + t_(2) + t_(3) + t_(4) is

Points A,B,C,D, on y^(2) = 4ax have parameters t_(1) , t_(2), t_(3) , t_(4) respectively . If AB bot CD then (t_(1) + t_(2)) (t_(3) + t_(4)) =

The point of intersection of tangents at 't'_(1) " and " 't'_(2) to the hyperbola xy = c^(2) is

Show that the normal to the rectangular hyperbola xy = c^(2) at the point t meets the curve again at a point t' such that t^(3)t' = - 1 .