A circle cuts the rectangular hyperbola `xy=1` in the points `(x_(1),y_(1)), r=1,2,3,4`.
Prove that `x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=1`

If a circle cuts the rectangular hyperbola xy = 1 in the points (x_(r), y_(r)), r = 1, 2, 3, 4 prove that x_(1), x_(2),x_(3), x_(4) = y_(1), y_(2), y_(3), y_(4) = 1

If the circle x^(2)+y^(2)=r^(2) intersects the hyperbola xy=c^(2) in four points (x_(i),y_(i)) for i=1,2,3 and 4 then y_(1)+y_(2)+y_(3)+y_(4)=

If the coordinates of the ends of a focal chord of the parabola y^(2)=4ax are (x_(1),y_(1)) "and" (x_(1),y_(2)) , then prove that x_(1)x_(2)=a^(2),y_(1)y_(2)=4a^(2) .

If the hyperbola xy=c^(2) intersects the circle x^(2)+y^(2)=a^(2)" is four points "P(x_(1),y_(1)), Q(x_(2),y_(2)), R(x_(3),y_(3)) and S(x_(4),y_(4)) then show that (i) x_(1)+x_(2)+x_(3)+x_(4)=0 (ii) y_(1)+y_(2)+y_(3)+y_(4)=0 (iii) x_(1)x_(2)x_(3)x_(4)=c^(4) (iv) y_(1)y_(2)y_(3)y_(4)=c^(4)

If the normal at (x_(i) y_(i)) i=1,2,3,4 on xy=c^2 meet at the point (alpha, beta) show that sum x_(i)=alpha, sum y_(i)=beta, sum x_(i)^(2)=alpha^(2), sum y^(2)=beta^(2), x_(1)x_(2)x_(3)x_(4)=y_(1)y_(2)y_(3)y_(4)=-c^(4)

If the ends of a focal chord of the parabola y^(2)=4ax are (x_(1),y_(1)) and (x_(2),y_(2)) then x_(1)x_(2)+y_(1)y_(2) =

If the tangent at (x_(0),y_(0)) to the curve x^(3)+y^(3)=a^(3) meet the curve again at (x_(1)y_(1)) prove that (x_(1))/(x_(0))+(y_(1))/(y_(0))=-1

If the circle x ^(2) +y^(2) =a^(2) intersects the hyperbola xy =c^(2) in four point (x_i,y_i) for i=1,2,3, and 4 then y_1+ y_2 +y_3 +y_4 =

At the point (x_(1),y_(1)) on the curve x^(3)+y^(3)=3axy show that the tangent is (x_(1)^(2)-ay_(1))x+(y_(1)^(2)-ax_(1))y=ax_(1)y_(1)